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Theorem dffi3 8887
 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 8879 . . . 4 (𝐴𝑉 → (fi‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
2 fr0g 8063 . . . . . . . 8 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3 frfnom 8062 . . . . . . . . 9 (rec(𝑅, 𝐴) ↾ ω) Fn ω
4 peano1 7593 . . . . . . . . 9 ∅ ∈ ω
5 fnfvelrn 6841 . . . . . . . . 9 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
63, 4, 5mp2an 690 . . . . . . . 8 ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω)
72, 6eqeltrrdi 2920 . . . . . . 7 (𝐴𝑉𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8 elssuni 4859 . . . . . . 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 3366 . . . . . . . . 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 4914 . . . . . . . . . 10 (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12 eliun 4914 . . . . . . . . . 10 (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
1311, 12anbi12i 628 . . . . . . . . 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 6998 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ran (rec(𝑅, 𝐴) ↾ ω))
1514eleq2d 2896 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ran (rec(𝑅, 𝐴) ↾ ω)))
16 fniunfv 6998 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ran (rec(𝑅, 𝐴) ↾ ω))
1716eleq2d 2896 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
1815, 17anbi12d 632 . . . . . . . . . 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 301 . . . . . . . 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 7581 . . . . . . . . . . . . . . . 16 Ord ω
22 ordunel 7534 . . . . . . . . . . . . . . . 16 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2321, 22mp3an1 1441 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2423adantl 484 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚𝑛) ∈ ω)
25 simprl 769 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
2624, 25jca 514 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27 nnon 7578 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ω → 𝑦 ∈ On)
28 nnon 7578 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ω → 𝑥 ∈ On)
2928ad2antlr 725 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
30 onsseleq 6225 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
3127, 29, 30syl2an2 684 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
32 rzal 4451 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
3332biantrud 534 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
34 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
3534sseq1d 3996 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
3633, 35bitr3d 283 . . . . . . . . . . . . . . . . . . . . . . . 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)))
37 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
3837sseq1d 3996 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
3937sseq2d 3997 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4039raleqbi1dv 3402 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
4138, 40anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . 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))))
42 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
4342sseq1d 3996 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
4442sseq2d 3997 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
4544raleqbi1dv 3402 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
4643, 45anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . 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))))
47 ssfii 8875 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
482, 47eqsstrd 4003 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴))
49 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
50 eqidd 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
51 ineq1 4179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎 = 𝑥 → (𝑎𝑏) = (𝑥𝑏))
5251eqeq2d 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑥 → (𝑥 = (𝑎𝑏) ↔ 𝑥 = (𝑥𝑏)))
53 ineq2 4181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑥 → (𝑥𝑏) = (𝑥𝑥))
54 inidm 4193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥𝑥) = 𝑥
5553, 54syl6eq 2870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = 𝑥 → (𝑥𝑏) = 𝑥)
5655eqeq2d 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑥 → (𝑥 = (𝑥𝑏) ↔ 𝑥 = 𝑥))
5752, 56rspc2ev 3633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
5849, 49, 50, 57syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
59 eqid 2819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
6059rnmpo 7276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)}
6160abeq2i 2946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
6258, 61sylibr 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
6362ssriv 3969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
64 simpl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
65 fvex 6676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6665uniex 7459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6766pwex 5272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
68 inss1 4203 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ⊆ 𝑎
69 elssuni 4859 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7069adantr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7168, 70sstrid 3976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
72 vex 3496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑎 ∈ V
7372inex1 5212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ∈ V
7473elpw 4544 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7571, 74sylibr 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7675rgen2 3201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
7759fmpo 7758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
7876, 77mpbi 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
79 frn 6513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
8078, 79ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
8167, 80ssexi 5217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V
82 nfcv 2975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝐴
83 nfcv 2975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝑛
84 nfcv 2975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
85 dffi3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
86 mpoeq12 7219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑢 = 𝑣𝑢 = 𝑣) → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
8786anidms 569 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
88 ineq1 4179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 = 𝑎 → (𝑦𝑧) = (𝑎𝑧))
89 ineq2 4181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑧 = 𝑏 → (𝑎𝑧) = (𝑎𝑏))
9088, 89cbvmpov 7241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))
9187, 90syl6eq 2870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9291rneqd 5801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑢 = 𝑣 → ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9392cbvmptv 5160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧))) = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9485, 93eqtri 2842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
95 rdgeq1 8039 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))) → rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴))
9694, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴)
9796reseq1i 5842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴) ↾ ω)
98 mpoeq12 7219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
9998anidms 569 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10099rneqd 5801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
10182, 83, 84, 97, 100frsucmpt 8065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
10264, 81, 101sylancl 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
10363, 102sseqtrrid 4018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
104 sstr2 3972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
105103, 104syl5com 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)))
106105ralimdv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
107 vex 3496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑛 ∈ V
108 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
109108sseq1d 3996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
110107, 109ralsn 4611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
111103, 110sylibr 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
112106, 111jctird 529 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
113 df-suc 6190 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 suc 𝑛 = (𝑛 ∪ {𝑛})
114113raleqi 3412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
115 ralunb 4165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
116114, 115bitri 277 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
117112, 116syl6ibr 254 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
118 fiin 8878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎𝑏) ∈ (fi‘𝐴))
119118rgen2 3201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴)
120 ss2ralv 4033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
121119, 120mpi 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴))
12259fmpo 7758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
123121, 122sylib 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
124123frnd 6514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
125124adantl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
126102, 125eqsstrd 4003 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
127117, 126jctild 528 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
128127expimpd 456 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
129128a1d 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)))))
13036, 41, 46, 48, 129finds2 7602 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ω → (𝐴𝑉 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
131130impcom 410 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝑥 ∈ ω) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
132131simprd 498 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
133132r19.21bi 3206 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
134133ex 415 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝑥 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
135134adantr 483 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
136 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
137 eqimss 4021 . . . . . . . . . . . . . . . . . . . 20 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
138136, 137syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
139138a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
140135, 139jaod 855 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝑦𝑥𝑦 = 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
14131, 140sylbid 242 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
142141ralrimiva 3180 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
143142ralrimiva 3180 . . . . . . . . . . . . . 14 (𝐴𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
144143adantr 483 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
145 ssun1 4146 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
146145a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚𝑛))
147 sseq2 3991 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (𝑦𝑥𝑦 ⊆ (𝑚𝑛)))
148 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
149148sseq2d 3997 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
150147, 149imbi12d 347 . . . . . . . . . . . . . 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)))))
151 sseq1 3990 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑚 ⊆ (𝑚𝑛)))
152 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
153152sseq1d 3996 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
154151, 153imbi12d 347 . . . . . . . . . . . . . 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)))))
155150, 154rspc2v 3631 . . . . . . . . . . . . 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)))))
15626, 144, 146, 155syl3c 66 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
157156sseld 3964 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
158 simprr 771 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ∈ ω)
15924, 158jca 514 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω))
160 ssun2 4147 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
161160a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚𝑛))
162 sseq1 3990 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑛 ⊆ (𝑚𝑛)))
163108sseq1d 3996 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
164162, 163imbi12d 347 . . . . . . . . . . . . . 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)))))
165150, 164rspc2v 3631 . . . . . . . . . . . . 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)))))
166159, 144, 161, 165syl3c 66 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
167166sseld 3964 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
16823ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑚𝑛) ∈ ω)
169 peano2 7594 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ ω → suc (𝑚𝑛) ∈ ω)
170 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑥 = suc (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
171170ssiun2s 4963 . . . . . . . . . . . . . . 15 (suc (𝑚𝑛) ∈ ω → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
172168, 169, 1713syl 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))
173 simprl 769 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
174 simprr 771 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
175 eqidd 2820 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) = (𝑐𝑑))
176 ineq1 4179 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
177176eqeq2d 2830 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ((𝑐𝑑) = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑐𝑏)))
178 ineq2 4181 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
179178eqeq2d 2830 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → ((𝑐𝑑) = (𝑐𝑏) ↔ (𝑐𝑑) = (𝑐𝑑)))
180177, 179rspc2ev 3633 . . . . . . . . . . . . . . . . . 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))
181173, 174, 175, 180syl3anc 1365 . . . . . . . . . . . . . . . . 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))
182 vex 3496 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
183182inex1 5212 . . . . . . . . . . . . . . . . . 18 (𝑐𝑑) ∈ V
184 eqeq1 2823 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑐𝑑) → (𝑥 = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑎𝑏)))
1851842rexbidv 3298 . . . . . . . . . . . . . . . . . 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)))
186183, 185elab 3665 . . . . . . . . . . . . . . . . 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))
187181, 186sylibr 236 . . . . . . . . . . . . . . . 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)})
188 eqid 2819 . . . . . . . . . . . . . . . . 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))
189188rnmpo 7276 . . . . . . . . . . . . . . . 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)}
190187, 189eleqtrrdi 2922 . . . . . . . . . . . . . . 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)))
191 fvex 6676 . . . . . . . . . . . . . . . . . . 19 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
192191uniex 7459 . . . . . . . . . . . . . . . . . 18 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
193192pwex 5272 . . . . . . . . . . . . . . . . 17 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
194 elssuni 4859 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19568, 194sstrid 3976 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19673elpw 4544 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
197195, 196sylibr 236 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
198197adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
199198rgen2 3201 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
200188fmpo 7758 . . . . . . . . . . . . . . . . . . 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)))
201199, 200mpbi 232 . . . . . . . . . . . . . . . . . 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))
202 frn 6513 . . . . . . . . . . . . . . . . . 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)))
203201, 202ax-mp 5 . . . . . . . . . . . . . . . . 17 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
204193, 203ssexi 5217 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V
205 nfcv 2975 . . . . . . . . . . . . . . . . 17 𝑣(𝑚𝑛)
206 nfcv 2975 . . . . . . . . . . . . . . . . 17 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
207 mpoeq12 7219 . . . . . . . . . . . . . . . . . . 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)))
208207anidms 569 . . . . . . . . . . . . . . . . . 18 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
209208rneqd 5801 . . . . . . . . . . . . . . . . 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)))
21082, 205, 206, 97, 209frsucmpt 8065 . . . . . . . . . . . . . . . 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)))
211168, 204, 210sylancl 588 . . . . . . . . . . . . . . 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)))
212190, 211eleqtrrd 2914 . . . . . . . . . . . . . 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)))
213172, 212sseldd 3966 . . . . . . . . . . . . 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))
214 fniunfv 6998 . . . . . . . . . . . . . 14 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω))
2153, 214ax-mp 5 . . . . . . . . . . . . 13 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω)
216213, 215eleqtrdi 2921 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
217216ex 415 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
218157, 167, 217syl2and 609 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
219218rexlimdvva 3292 . . . . . . . . 9 (𝐴𝑉 → (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
220219imp 409 . . . . . . . 8 ((𝐴𝑉 ∧ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
22120, 220sylan2br 596 . . . . . . 7 ((𝐴𝑉 ∧ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
222221ralrimivva 3189 . . . . . 6 (𝐴𝑉 → ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
223131simpld 497 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
224 fvex 6676 . . . . . . . . . . . . 13 (fi‘𝐴) ∈ V
225224elpw2 5239 . . . . . . . . . . . 12 (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
226223, 225sylibr 236 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
227226ralrimiva 3180 . . . . . . . . . 10 (𝐴𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
228 fnfvrnss 6877 . . . . . . . . . 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))
2293, 227, 228sylancr 589 . . . . . . . . 9 (𝐴𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
230 sspwuni 5013 . . . . . . . . 9 (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
231229, 230sylib 220 . . . . . . . 8 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
232 ssexg 5218 . . . . . . . 8 (( ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
233231, 224, 232sylancl 588 . . . . . . 7 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
234 sseq2 3991 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴𝑥𝐴 ran (rec(𝑅, 𝐴) ↾ ω)))
235 eleq2 2899 . . . . . . . . . . 11 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐𝑑) ∈ 𝑥 ↔ (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
236235raleqbi1dv 3402 . . . . . . . . . 10 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
237236raleqbi1dv 3402 . . . . . . . . 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)))
238234, 237anbi12d 632 . . . . . . . 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))))
239238elabg 3664 . . . . . . 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))))
240233, 239syl 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))))
2419, 222, 240mpbir2and 711 . . . . 5 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
242 intss1 4882 . . . . 5 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
243241, 242syl 17 . . . 4 (𝐴𝑉 {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
2441, 243eqsstrd 4003 . . 3 (𝐴𝑉 → (fi‘𝐴) ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
245244, 231eqssd 3982 . 2 (𝐴𝑉 → (fi‘𝐴) = ran (rec(𝑅, 𝐴) ↾ ω))
246 df-ima 5561 . . 3 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
247246unieqi 4839 . 2 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
248245, 247syl6eqr 2872 1 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1530   ∈ wcel 2107  {cab 2797  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ∪ cun 3932   ∩ cin 3933   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  {csn 4559  ∪ cuni 4830  ∩ cint 4867  ∪ ciun 4910   ↦ cmpt 5137   × cxp 5546  ran crn 5549   ↾ cres 5550   “ cima 5551  Ord word 6183  Oncon0 6184  suc csuc 6186   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348   ∈ cmpo 7150  ωcom 7572  reccrdg 8037  ficfi 8866 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-fin 8505  df-fi 8867 This theorem is referenced by: (None)
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