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Theorem dffi3 9367
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 9359 . . . 4 (𝐴𝑉 → (fi‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
2 fr0g 8382 . . . . . . . 8 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3 frfnom 8381 . . . . . . . . 9 (rec(𝑅, 𝐴) ↾ ω) Fn ω
4 peano1 7825 . . . . . . . . 9 ∅ ∈ ω
5 fnfvelrn 7031 . . . . . . . . 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 2847 . . . . . . 7 (𝐴𝑉𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8 elssuni 4898 . . . . . . 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 3217 . . . . . . . . 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 4958 . . . . . . . . . 10 (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12 eliun 4958 . . . . . . . . . 10 (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
1311, 12anbi12i 627 . . . . . . . . 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 7194 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ran (rec(𝑅, 𝐴) ↾ ω))
1514eleq2d 2823 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ran (rec(𝑅, 𝐴) ↾ ω)))
16 fniunfv 7194 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ran (rec(𝑅, 𝐴) ↾ ω))
1716eleq2d 2823 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
1815, 17anbi12d 631 . . . . . . . . . 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 298 . . . . . . . 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 7812 . . . . . . . . . . . . . . . 16 Ord ω
22 ordunel 7762 . . . . . . . . . . . . . . . 16 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2321, 22mp3an1 1448 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2423adantl 482 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚𝑛) ∈ ω)
25 simprl 769 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
2624, 25jca 512 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27 nnon 7808 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ω → 𝑦 ∈ On)
28 nnon 7808 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ω → 𝑥 ∈ On)
2928ad2antlr 725 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
30 onsseleq 6358 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
3127, 29, 30syl2an2 684 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
32 rzal 4466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
3332biantrud 532 . . . . . . . . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
3534sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
3633, 35bitr3d 280 . . . . . . . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
3837sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
3937sseq2d 3976 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4039raleqbi1dv 3307 . . . . . . . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
4342sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
4442sseq2d 3976 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3307 . . . . . . . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . . . . . . . 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 9355 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
482, 47eqsstrd 3982 . . . . . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
51 ineq1 4165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎 = 𝑥 → (𝑎𝑏) = (𝑥𝑏))
5251eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑥 → (𝑥 = (𝑎𝑏) ↔ 𝑥 = (𝑥𝑏)))
53 ineq2 4166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑥 → (𝑥𝑏) = (𝑥𝑥))
54 inidm 4178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥𝑥) = 𝑥
5553, 54eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = 𝑥 → (𝑥𝑏) = 𝑥)
5655eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑥 → (𝑥 = (𝑥𝑏) ↔ 𝑥 = 𝑥))
5752, 56rspc2ev 3592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
59 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)}
6160eqabi 2881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
6362ssriv 3948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
64 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
65 fvex 6855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6665uniex 7678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6766pwex 5335 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
68 inss1 4188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ⊆ 𝑎
69 elssuni 4898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7069adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7168, 70sstrid 3955 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
72 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑎 ∈ V
7372inex1 5274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ∈ V
7473elpw 4564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7571, 74sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7675rgen2 3194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
7759fmpo 8000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V
82 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝐴
83 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝑛
84 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑢 = 𝑣𝑢 = 𝑣) → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
8786anidms 567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
88 ineq1 4165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 = 𝑎 → (𝑦𝑧) = (𝑎𝑧))
89 ineq2 4166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑧 = 𝑏 → (𝑎𝑧) = (𝑎𝑏))
9088, 89cbvmpov 7452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))
9187, 90eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9291rneqd 5893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑢 = 𝑣 → ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9392cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧))) = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9485, 93eqtri 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
95 rdgeq1 8357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5933 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴) ↾ ω)
98 mpoeq12 7430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10099rneqd 5893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
104 sstr2 3951 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑛 ∈ V
108 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
109108sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 527 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 suc 𝑛 = (𝑛 ∪ {𝑛})
114113raleqi 3311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 251 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎𝑏) ∈ (fi‘𝐴))
119118rgen2 3194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴)
120 ss2ralv 4012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
127117, 126jctild 526 . . . . . . . . . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . . . . . . . . . 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 7837 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ω → (𝐴𝑉 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
131130impcom 408 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝑥 ∈ ω) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
132131simprd 496 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
133132r19.21bi 3234 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
134133ex 413 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝑥 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
135134adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
136 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
137 eqimss 4000 . . . . . . . . . . . . . . . . . . . 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 857 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝑦𝑥𝑦 = 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
14131, 140sylbid 239 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
142141ralrimiva 3143 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
143142ralrimiva 3143 . . . . . . . . . . . . . 14 (𝐴𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
144143adantr 481 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
145 ssun1 4132 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
146145a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚𝑛))
147 sseq2 3970 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (𝑦𝑥𝑦 ⊆ (𝑚𝑛)))
148 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
149148sseq2d 3976 . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑚 ⊆ (𝑚𝑛)))
152 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
153152sseq1d 3975 . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . 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 3590 . . . . . . . . . . . . 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 3943 . . . . . . . . . . 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 512 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω))
160 ssun2 4133 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
161160a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚𝑛))
162 sseq1 3969 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑛 ⊆ (𝑚𝑛)))
163108sseq1d 3975 . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . 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 3590 . . . . . . . . . . . . 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 3943 . . . . . . . . . . 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 7827 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ ω → suc (𝑚𝑛) ∈ ω)
170 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑥 = suc (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
171170ssiun2s 5008 . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) = (𝑐𝑑))
176 ineq1 4165 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
177176eqeq2d 2747 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ((𝑐𝑑) = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑐𝑏)))
178 ineq2 4166 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
179178eqeq2d 2747 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → ((𝑐𝑑) = (𝑐𝑏) ↔ (𝑐𝑑) = (𝑐𝑑)))
180177, 179rspc2ev 3592 . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
183182inex1 5274 . . . . . . . . . . . . . . . . . 18 (𝑐𝑑) ∈ V
184 eqeq1 2740 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑐𝑑) → (𝑥 = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑎𝑏)))
1851842rexbidv 3213 . . . . . . . . . . . . . . . . . 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 3630 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . 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 7489 . . . . . . . . . . . . . . . 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 2849 . . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . . . . . 19 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
192191uniex 7678 . . . . . . . . . . . . . . . . . 18 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
193192pwex 5335 . . . . . . . . . . . . . . . . 17 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
194 elssuni 4898 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19568, 194sstrid 3955 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19673elpw 4564 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
197195, 196sylibr 233 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
198197adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
199198rgen2 3194 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
200188fmpo 8000 . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . . 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 6675 . . . . . . . . . . . . . . . . . 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 5279 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V
205 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑣(𝑚𝑛)
206 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
207 mpoeq12 7430 . . . . . . . . . . . . . . . . . . 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 567 . . . . . . . . . . . . . . . . . 18 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
209208rneqd 5893 . . . . . . . . . . . . . . . . 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 8384 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . 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 7194 . . . . . . . . . . . . . 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 2848 . . . . . . . . . . . 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 413 . . . . . . . . . . 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 608 . . . . . . . . . 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 3205 . . . . . . . . 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 407 . . . . . . . 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 595 . . . . . . 7 ((𝐴𝑉 ∧ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
222221ralrimivva 3197 . . . . . 6 (𝐴𝑉 → ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
223131simpld 495 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
224 fvex 6855 . . . . . . . . . . . . 13 (fi‘𝐴) ∈ V
225224elpw2 5302 . . . . . . . . . . . 12 (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
226223, 225sylibr 233 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
227226ralrimiva 3143 . . . . . . . . . 10 (𝐴𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
228 fnfvrnss 7068 . . . . . . . . . 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 587 . . . . . . . . 9 (𝐴𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
230 sspwuni 5060 . . . . . . . . 9 (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
231229, 230sylib 217 . . . . . . . 8 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
232 ssexg 5280 . . . . . . . 8 (( ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
233231, 224, 232sylancl 586 . . . . . . 7 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
234 sseq2 3970 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴𝑥𝐴 ran (rec(𝑅, 𝐴) ↾ ω)))
235 eleq2 2826 . . . . . . . . . . 11 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐𝑑) ∈ 𝑥 ↔ (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
236235raleqbi1dv 3307 . . . . . . . . . 10 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
237236raleqbi1dv 3307 . . . . . . . . 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 631 . . . . . . . 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 3628 . . . . . . 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 4924 . . . . 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 3982 . . 3 (𝐴𝑉 → (fi‘𝐴) ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
245244, 231eqssd 3961 . 2 (𝐴𝑉 → (fi‘𝐴) = ran (rec(𝑅, 𝐴) ↾ ω))
246 df-ima 5646 . . 3 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
247246unieqi 4878 . 2 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
248245, 247eqtr4di 2794 1 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865   cint 4907   ciun 4954  cmpt 5188   × cxp 5631  ran crn 5634  cres 5635  cima 5636  Ord word 6316  Oncon0 6317  suc csuc 6319   Fn wfn 6491  wf 6492  cfv 6496  cmpo 7359  ωcom 7802  reccrdg 8355  ficfi 9346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-fin 8887  df-fi 9347
This theorem is referenced by: (None)
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