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Theorem fpr3g 32733
Description: Functions defined by founded recursion over a partial ordering are identical up to relation, domain, and characteristic function. This version of frr3g 32732 does not require infinity. (Contributed by Scott Fenton, 24-Aug-2022.)
Assertion
Ref Expression
fpr3g (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝑦,𝐻   𝑦,𝑅

Proof of Theorem fpr3g
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2798 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐴 = 𝐴)
2 r19.21v 3144 . . . . . 6 (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤)) ↔ (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
3 simprll 775 . . . . . . . . . . . 12 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))) → 𝐹 Fn 𝐴)
4 simprrl 777 . . . . . . . . . . . 12 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))) → 𝐺 Fn 𝐴)
5 predss 6037 . . . . . . . . . . . . 13 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
6 fvreseq 6682 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
75, 6mpan2 687 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
83, 4, 7syl2anc 584 . . . . . . . . . . 11 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
98biimp3ar 1462 . . . . . . . . . 10 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
109oveq2d 7039 . . . . . . . . 9 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
11 fveq2 6545 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
12 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑦 = 𝑧)
13 predeq3 6034 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
1413reseq2d 5741 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
1512, 14oveq12d 7041 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
1611, 15eqeq12d 2812 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
17 simp2lr 1234 . . . . . . . . . 10 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
18 simp1 1129 . . . . . . . . . 10 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → 𝑧𝐴)
1916, 17, 18rspcdva 3567 . . . . . . . . 9 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
20 fveq2 6545 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐺𝑦) = (𝐺𝑧))
2113reseq2d 5741 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
2212, 21oveq12d 7041 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
2320, 22eqeq12d 2812 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
24 simp2rr 1236 . . . . . . . . . 10 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))
2523, 24, 18rspcdva 3567 . . . . . . . . 9 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
2610, 19, 253eqtr4d 2843 . . . . . . . 8 ((𝑧𝐴 ∧ ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹𝑧) = (𝐺𝑧))
27263exp 1112 . . . . . . 7 (𝑧𝐴 → (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
2827a2d 29 . . . . . 6 (𝑧𝐴 → ((((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
292, 28syl5bi 243 . . . . 5 (𝑧𝐴 → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
30 fveq2 6545 . . . . . . 7 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
31 fveq2 6545 . . . . . . 7 (𝑧 = 𝑤 → (𝐺𝑧) = (𝐺𝑤))
3230, 31eqeq12d 2812 . . . . . 6 (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑤) = (𝐺𝑤)))
3332imbi2d 342 . . . . 5 (𝑧 = 𝑤 → ((((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)) ↔ (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤))))
3429, 33frpoins2g 32694 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)))
35 r19.21v 3144 . . . 4 (∀𝑧𝐴 (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)) ↔ (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
3634, 35sylib 219 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
37363impib 1109 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))
38 simp2l 1192 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 Fn 𝐴)
39 simp3l 1194 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐺 Fn 𝐴)
40 eqfnfv2 6675 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
4138, 39, 40syl2anc 584 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
421, 37, 41mpbir2and 709 1 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wral 3107  wss 3865   Po wpo 5367   Fr wfr 5406   Se wse 5407  cres 5452  Predcpred 6029   Fn wfn 6227  cfv 6232  (class class class)co 7023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-po 5369  df-fr 5409  df-se 5410  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-iota 6196  df-fun 6234  df-fn 6235  df-fv 6240  df-ov 7026
This theorem is referenced by:  fprlem1  32748  fpr3  32752
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