Theorem frr3g 9748

Description: Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9753. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
frr3g	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝑦,𝐻   𝑦,𝑅

Proof of Theorem frr3g

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ra4v 3872	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)))
2		r19.26 3103	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3	2	anbi2i 622	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))))
4		fveq2 6882	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
5		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
6		predeq3 6295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
7	6	reseq2d 5972	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
8	5, 7	oveq12d 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9	4, 8	eqeq12d 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
10		fveq2 6882	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧))
11	6	reseq2d 5972	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
12	5, 11	oveq12d 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
13	10, 12	eqeq12d 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
14	9, 13	anbi12d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))))
15	14	rspcva 3602	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
16		eqtr3 2750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐹‘𝑧))
17	16	eqcomd 2730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
18		eqtr3 2750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧))
19	18	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → ((𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
20	17, 19	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
21	20	expimpd 453	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) → (((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
22		predss 6299	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
23		fvreseq 7032	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)))
24	22, 23	mpan2 688	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)))
25	24	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
26	25	oveq2d 7418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
27	26	eqcomd 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
28	21, 27	syl11 33	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐹‘𝑧) = (𝐺‘𝑧)))
29	28	expd 415	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
30	15, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
31	30	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))))
32	31	com23 86	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))))
33	32	impd 410	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
34	3, 33	biimtrrid 242	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
35	34	a2d 29	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))))
36	1, 35	syl5 34	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))))
37		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
38		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
39	37, 38	eqeq12d 2740	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑤) = (𝐺‘𝑤)))
40	39	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)) ↔ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤))))
41	36, 40	frins2 9746	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
42		rsp 3236	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)) → (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))))
43	41, 42	syl 17	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))))
44	43	com3r 87	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧))))
45	44	an4s 657	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧))))
46	45	com12 32	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧))))
47	46	3impib 1113	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧)))
48	47	ralrimiv 3137	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))
49		eqid 2724	. . 3 ⊢ 𝐴 = 𝐴
50	48, 49	jctil 519	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
51		eqfnfv2 7024	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
52	51	ad2ant2r 744	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
53	52	3adant1 1127	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
54	50, 53	mpbird 257	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053   ⊆ wss 3941   Fr wfr 5619   Se wse 5620   ↾ cres 5669  Predcpred 6290   Fn wfn 6529  ‘cfv 6534  (class class class)co 7402

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-ttrcl 9700

This theorem is referenced by:  frrlem15  9749  frr3  9753