Theorem frr3g 9680

Description: Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9685. (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 3837	. . . . . . . . . . 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 3098	. . . . . . . . . . . . . 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 624	. . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
5		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
6		predeq3 6271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
7	6	reseq2d 5946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
8	5, 7	oveq12d 7386	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9	4, 8	eqeq12d 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
10		fveq2 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧))
11	6	reseq2d 5946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
12	5, 11	oveq12d 7386	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
13	10, 12	eqeq12d 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
14	9, 13	anbi12d 633	. . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . . . . . . 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 6275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
23		fvreseq 6994	. . . . . . . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
38		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
39	37, 38	eqeq12d 2753	. . . . . . . . . . 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 9678	. . . . . . . . 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 3226	. . . . . . . . 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 661	. . . . . 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 1117	. . . 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 3129	. . 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 2737	. . 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 6986	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
52	51	ad2ant2r 748	. . 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 1131	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903   Fr wfr 5582   Se wse 5583   ↾ cres 5634  Predcpred 6266   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-ttrcl 9629

This theorem is referenced by:  frrlem15  9681  frr3  9685