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Theorem dffi3 8573
Description: The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
dffi3.1 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
Assertion
Ref Expression
dffi3 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑉   𝑦,𝑢,𝑧
Allowed substitution hints:   𝐴(𝑧,𝑢)   𝑅(𝑧,𝑢)   𝑉(𝑧,𝑢)

Proof of Theorem dffi3
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑚 𝑛 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffi2 8565 . . . 4 (𝐴𝑉 → (fi‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
2 fr0g 7764 . . . . . . . 8 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3 frfnom 7763 . . . . . . . . 9 (rec(𝑅, 𝐴) ↾ ω) Fn ω
4 peano1 7312 . . . . . . . . 9 ∅ ∈ ω
5 fnfvelrn 6575 . . . . . . . . 9 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
63, 4, 5mp2an 675 . . . . . . . 8 ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω)
72, 6syl6eqelr 2893 . . . . . . 7 (𝐴𝑉𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8 elssuni 4657 . . . . . . 7 (𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω) → 𝐴 ran (rec(𝑅, 𝐴) ↾ ω))
97, 8syl 17 . . . . . 6 (𝐴𝑉𝐴 ran (rec(𝑅, 𝐴) ↾ ω))
10 reeanv 3294 . . . . . . . . 9 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
11 eliun 4712 . . . . . . . . . 10 (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12 eliun 4712 . . . . . . . . . 10 (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
1311, 12anbi12i 614 . . . . . . . . 9 ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
14 fniunfv 6726 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ran (rec(𝑅, 𝐴) ↾ ω))
1514eleq2d 2870 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ran (rec(𝑅, 𝐴) ↾ ω)))
16 fniunfv 6726 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ran (rec(𝑅, 𝐴) ↾ ω))
1716eleq2d 2870 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
1815, 17anbi12d 618 . . . . . . . . . 10 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))))
193, 18ax-mp 5 . . . . . . . . 9 ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
2010, 13, 193bitr2i 290 . . . . . . . 8 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
21 ordom 7301 . . . . . . . . . . . . . . . 16 Ord ω
22 ordunel 7254 . . . . . . . . . . . . . . . 16 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2321, 22mp3an1 1565 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2423adantl 469 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚𝑛) ∈ ω)
25 simprl 778 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
2624, 25jca 503 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27 nnon 7298 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ω → 𝑦 ∈ On)
2827adantl 469 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
29 nnon 7298 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ω → 𝑥 ∈ On)
3029ad2antlr 709 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
31 onsseleq 5975 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
3228, 30, 31syl2anc 575 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
33 rzal 4265 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
3433biantrud 523 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
35 fveq2 6405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
3635sseq1d 3826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
3734, 36bitr3d 272 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ∅ → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
38 fveq2 6405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
3938sseq1d 3826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
4038sseq2d 3827 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4140raleqbi1dv 3334 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4239, 41anbi12d 618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))))
43 fveq2 6405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
4443sseq1d 3826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
4543sseq2d 3827 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
4645raleqbi1dv 3334 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
4744, 46anbi12d 618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = suc 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
48 ssfii 8561 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
492, 48eqsstrd 3833 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴))
50 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
51 eqidd 2806 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
52 ineq1 4003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎 = 𝑥 → (𝑎𝑏) = (𝑥𝑏))
5352eqeq2d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑥 → (𝑥 = (𝑎𝑏) ↔ 𝑥 = (𝑥𝑏)))
54 ineq2 4004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑥 → (𝑥𝑏) = (𝑥𝑥))
55 inidm 4016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥𝑥) = 𝑥
5654, 55syl6eq 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = 𝑥 → (𝑥𝑏) = 𝑥)
5756eqeq2d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑥 → (𝑥 = (𝑥𝑏) ↔ 𝑥 = 𝑥))
5853, 57rspc2ev 3516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 = 𝑥) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
5950, 50, 51, 58syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
60 eqid 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
6160rnmpt2 6997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏)}
6261abeq2i 2918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
6359, 62sylibr 225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
6463ssriv 3799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
65 simpl 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
66 fvex 6418 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6766uniex 7180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6867pwex 5047 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
69 inss1 4026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ⊆ 𝑎
70 elssuni 4657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7170adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7269, 71syl5ss 3806 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
73 vex 3393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑎 ∈ V
7473inex1 4991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ∈ V
7574elpw 4354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7672, 75sylibr 225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7776rgen2a 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
7860fmpt2 7467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7977, 78mpbi 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
80 frn 6259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
8179, 80ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
8268, 81ssexi 4995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V
83 nfcv 2947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝐴
84 nfcv 2947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝑛
85 nfcv 2947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
86 dffi3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
87 mpt2eq12 6942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑢 = 𝑣𝑢 = 𝑣) → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
8887anidms 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
89 ineq1 4003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 = 𝑎 → (𝑦𝑧) = (𝑎𝑧))
90 ineq2 4004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑧 = 𝑏 → (𝑎𝑧) = (𝑎𝑏))
9189, 90cbvmpt2v 6962 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))
9288, 91syl6eq 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9392rneqd 5551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑢 = 𝑣 → ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9493cbvmptv 4940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧))) = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9586, 94eqtri 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
96 rdgeq1 7740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))) → rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴))
9795, 96ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴)
9897reseq1i 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴) ↾ ω)
99 mpt2eq12 6942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10099anidms 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
101100rneqd 5551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10283, 84, 85, 98, 101frsucmpt 7766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10365, 82, 102sylancl 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10464, 103syl5sseqr 3848 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
105 sstr2 3802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
106104, 105syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
107106ralimdv 3150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
108 vex 3393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑛 ∈ V
109 fveq2 6405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
110109sseq1d 3826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
111108, 110ralsn 4412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
112104, 111sylibr 225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
113107, 112jctird 518 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
114 df-suc 5939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 suc 𝑛 = (𝑛 ∪ {𝑛})
115114raleqi 3330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
116 ralunb 3990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
117115, 116bitri 266 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
118113, 117syl6ibr 243 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
119 fiin 8564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎𝑏) ∈ (fi‘𝐴))
120119rgen2a 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴)
121 ssralv 3860 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
122121ralimdv 3150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
123 ssralv 3860 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
124122, 123syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
125120, 124mpi 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴))
12660fmpt2 7467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
127125, 126sylib 209 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
128127frnd 6260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
129128adantl 469 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
130103, 129eqsstrd 3833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
131118, 130jctild 517 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
132131expimpd 443 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ω → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
133132a1d 25 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → (𝐴𝑉 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))))
13437, 42, 47, 49, 133finds2 7321 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ω → (𝐴𝑉 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
135134impcom 396 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝑥 ∈ ω) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
136135simprd 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
137136r19.21bi 3119 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
138137ex 399 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝑥 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
139138adantr 468 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
140 fveq2 6405 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
141 eqimss 3851 . . . . . . . . . . . . . . . . . . . 20 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
142140, 141syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
143142a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
144139, 143jaod 877 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝑦𝑥𝑦 = 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
14532, 144sylbid 231 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
146145ralrimiva 3153 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
147146ralrimiva 3153 . . . . . . . . . . . . . 14 (𝐴𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
148147adantr 468 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
149 ssun1 3972 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
150149a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚𝑛))
151 sseq2 3821 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (𝑦𝑥𝑦 ⊆ (𝑚𝑛)))
152 fveq2 6405 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
153152sseq2d 3827 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
154151, 153imbi12d 335 . . . . . . . . . . . . . 14 (𝑥 = (𝑚𝑛) → ((𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
155 sseq1 3820 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑚 ⊆ (𝑚𝑛)))
156 fveq2 6405 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
157156sseq1d 3826 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
158155, 157imbi12d 335 . . . . . . . . . . . . . 14 (𝑦 = 𝑚 → ((𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) ↔ (𝑚 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
159154, 158rspc2v 3514 . . . . . . . . . . . . 13 (((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑚 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
16026, 148, 150, 159syl3c 66 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
161160sseld 3794 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
162 simprr 780 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ∈ ω)
16324, 162jca 503 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω))
164 ssun2 3973 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
165164a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚𝑛))
166 sseq1 3820 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑛 ⊆ (𝑚𝑛)))
167109sseq1d 3826 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
168166, 167imbi12d 335 . . . . . . . . . . . . . 14 (𝑦 = 𝑛 → ((𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) ↔ (𝑛 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
169154, 168rspc2v 3514 . . . . . . . . . . . . 13 (((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑛 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
170163, 148, 165, 169syl3c 66 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
171170sseld 3794 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
17223ad2antlr 709 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑚𝑛) ∈ ω)
173 peano2 7313 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ ω → suc (𝑚𝑛) ∈ ω)
174 fveq2 6405 . . . . . . . . . . . . . . . 16 (𝑥 = suc (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
175174ssiun2s 4752 . . . . . . . . . . . . . . 15 (suc (𝑚𝑛) ∈ ω → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
176172, 173, 1753syl 18 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
177 simprl 778 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
178 simprr 780 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
179 eqidd 2806 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) = (𝑐𝑑))
180 ineq1 4003 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
181180eqeq2d 2815 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ((𝑐𝑑) = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑐𝑏)))
182 ineq2 4004 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
183182eqeq2d 2815 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → ((𝑐𝑑) = (𝑐𝑏) ↔ (𝑐𝑑) = (𝑐𝑑)))
184181, 183rspc2ev 3516 . . . . . . . . . . . . . . . . . 18 ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ (𝑐𝑑) = (𝑐𝑑)) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
185177, 178, 179, 184syl3anc 1483 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
186 vex 3393 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
187186inex1 4991 . . . . . . . . . . . . . . . . . 18 (𝑐𝑑) ∈ V
188 eqeq1 2809 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑐𝑑) → (𝑥 = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑎𝑏)))
1891882rexbidv 3244 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑐𝑑) → (∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏)))
190187, 189elab 3544 . . . . . . . . . . . . . . . . 17 ((𝑐𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)} ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
191185, 190sylibr 225 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)})
192 eqid 2805 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
193192rnmpt2 6997 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)}
194191, 193syl6eleqr 2895 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
195 fvex 6418 . . . . . . . . . . . . . . . . . . 19 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
196195uniex 7180 . . . . . . . . . . . . . . . . . 18 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
197196pwex 5047 . . . . . . . . . . . . . . . . 17 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
198 elssuni 4657 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19969, 198syl5ss 3806 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
20074elpw 4354 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
201199, 200sylibr 225 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
202201adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
203202rgen2a 3164 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
204192fmpt2 7467 . . . . . . . . . . . . . . . . . . 19 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
205203, 204mpbi 221 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
206 frn 6259 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
207205, 206ax-mp 5 . . . . . . . . . . . . . . . . 17 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
208197, 207ssexi 4995 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V
209 nfcv 2947 . . . . . . . . . . . . . . . . 17 𝑣(𝑚𝑛)
210 nfcv 2947 . . . . . . . . . . . . . . . . 17 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
211 mpt2eq12 6942 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
212211anidms 558 . . . . . . . . . . . . . . . . . 18 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
213212rneqd 5551 . . . . . . . . . . . . . . . . 17 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
21483, 209, 210, 98, 213frsucmpt 7766 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
215172, 208, 214sylancl 576 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
216194, 215eleqtrrd 2887 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
217176, 216sseldd 3796 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
218 fniunfv 6726 . . . . . . . . . . . . . 14 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω))
2193, 218ax-mp 5 . . . . . . . . . . . . 13 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω)
220217, 219syl6eleq 2894 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
221220ex 399 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
222161, 171, 221syl2and 597 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
223222rexlimdvva 3225 . . . . . . . . 9 (𝐴𝑉 → (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
224223imp 395 . . . . . . . 8 ((𝐴𝑉 ∧ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
22520, 224sylan2br 584 . . . . . . 7 ((𝐴𝑉 ∧ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
226225ralrimivva 3158 . . . . . 6 (𝐴𝑉 → ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
227135simpld 484 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
228 fvex 6418 . . . . . . . . . . . . 13 (fi‘𝐴) ∈ V
229228elpw2 5017 . . . . . . . . . . . 12 (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
230227, 229sylibr 225 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
231230ralrimiva 3153 . . . . . . . . . 10 (𝐴𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
232 fnfvrnss 6609 . . . . . . . . . 10 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴)) → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
2333, 231, 232sylancr 577 . . . . . . . . 9 (𝐴𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
234 sspwuni 4799 . . . . . . . . 9 (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
235233, 234sylib 209 . . . . . . . 8 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
236 ssexg 4996 . . . . . . . 8 (( ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
237235, 228, 236sylancl 576 . . . . . . 7 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
238 sseq2 3821 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴𝑥𝐴 ran (rec(𝑅, 𝐴) ↾ ω)))
239 eleq2 2873 . . . . . . . . . . 11 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐𝑑) ∈ 𝑥 ↔ (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
240239raleqbi1dv 3334 . . . . . . . . . 10 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
241240raleqbi1dv 3334 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
242238, 241anbi12d 618 . . . . . . . 8 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥) ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
243242elabg 3545 . . . . . . 7 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ V → ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
244237, 243syl 17 . . . . . 6 (𝐴𝑉 → ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
2459, 226, 244mpbir2and 695 . . . . 5 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
246 intss1 4680 . . . . 5 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
247245, 246syl 17 . . . 4 (𝐴𝑉 {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
2481, 247eqsstrd 3833 . . 3 (𝐴𝑉 → (fi‘𝐴) ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
249248, 235eqssd 3812 . 2 (𝐴𝑉 → (fi‘𝐴) = ran (rec(𝑅, 𝐴) ↾ ω))
250 df-ima 5321 . . 3 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
251250unieqi 4635 . 2 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
252249, 251syl6eqr 2857 1 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2158  {cab 2791  wral 3095  wrex 3096  Vcvv 3390  cun 3764  cin 3765  wss 3766  c0 4113  𝒫 cpw 4348  {csn 4367   cuni 4626   cint 4665   ciun 4708  cmpt 4919   × cxp 5306  ran crn 5309  cres 5310  cima 5311  Ord word 5932  Oncon0 5933  suc csuc 5935   Fn wfn 6093  wf 6094  cfv 6098  cmpt2 6873  ωcom 7292  reccrdg 7738  ficfi 8552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-en 8190  df-fin 8193  df-fi 8553
This theorem is referenced by: (None)
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