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Theorem fpr3g 8309
Description: Functions defined by well-founded recursion over a partial order are identical up to relation, domain, and characteristic function. This version of frr3g 9794 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 2736 . 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 3178 . . . . . 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 779 . . . . . . . . . . . 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 781 . . . . . . . . . . . 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 6331 . . . . . . . . . . . . 13 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
6 fvreseq 7060 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 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 1469 . . . . . . . . . 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 7447 . . . . . . . . 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 6907 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
12 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑦 = 𝑧)
13 predeq3 6327 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
1413reseq2d 6000 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
1512, 14oveq12d 7449 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
1611, 15eqeq12d 2751 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
17 simp2lr 1240 . . . . . . . . . 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 1135 . . . . . . . . . 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 3623 . . . . . . . . 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 6907 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐺𝑦) = (𝐺𝑧))
2113reseq2d 6000 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
2212, 21oveq12d 7449 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
2320, 22eqeq12d 2751 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
24 simp2rr 1242 . . . . . . . . . 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 3623 . . . . . . . . 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 2785 . . . . . . . 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 1118 . . . . . . 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, 28biimtrid 242 . . . . 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 6907 . . . . . . 7 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
31 fveq2 6907 . . . . . . 7 (𝑧 = 𝑤 → (𝐺𝑧) = (𝐺𝑤))
3230, 31eqeq12d 2751 . . . . . 6 (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑤) = (𝐺𝑤)))
3332imbi2d 340 . . . . 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 6368 . . . 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 3178 . . . 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 218 . . 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 1115 . 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 1198 . . 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 1200 . . 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 7052 . . 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 713 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963   Po wpo 5595   Fr wfr 5638   Se wse 5639  cres 5691  Predcpred 6322   Fn wfn 6558  cfv 6563  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434
This theorem is referenced by:  fprlem1  8324  fpr3  8329
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