Theorem foeqcnvco 6827

Description: Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)

Assertion

Ref	Expression
foeqcnvco	⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))

Proof of Theorem foeqcnvco

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fococnv2 6416	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))
2		cnveq 5541	. . . . . 6 ⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺)
3	2	coeq2d 5530	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐹) = (𝐹 ∘ ◡𝐺))
4	3	eqeq1d 2779	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵) ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
5	1, 4	syl5ibcom 237	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
6	5	adantr 474	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
7		fofn 6368	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
8	7	ad2antrr 716	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9		fofn 6368	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺 Fn 𝐴)
10	9	ad2antlr 717	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
11	9	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺 Fn 𝐴)
12		fnopfv 6615	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
13	11, 12	sylan 575	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
14		fvex 6459	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑥) ∈ V
15		vex 3400	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5550	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥))
17		df-br 4887	. . . . . . . . . . . 12 ⊢ (𝑥𝐺(𝐺‘𝑥) ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
18	16, 17	bitri 267	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
19	13, 18	sylibr 226	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)◡𝐺𝑥)
20	7	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹 Fn 𝐴)
21		fnopfv 6615	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
22	20, 21	sylan 575	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
23		df-br 4887	. . . . . . . . . . 11 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
24	22, 23	sylibr 226	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
25		breq2 4890	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐺‘𝑥)◡𝐺𝑦 ↔ (𝐺‘𝑥)◡𝐺𝑥))
26		breq1 4889	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦𝐹(𝐹‘𝑥) ↔ 𝑥𝐹(𝐹‘𝑥)))
27	25, 26	anbi12d 624	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)) ↔ ((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥))))
28	15, 27	spcev 3501	. . . . . . . . . 10 ⊢ (((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
29	19, 24, 28	syl2anc 579	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
30		fvex 6459	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
31	14, 30	brco 5538	. . . . . . . . 9 ⊢ ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
32	29, 31	sylibr 226	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
33	32	adantlr 705	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
34		breq 4888	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)))
35	34	ad2antlr 717	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)))
36	33, 35	mpbid 224	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))
37		fof 6366	. . . . . . . . . 10 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
38	37	adantl 475	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴⟶𝐵)
39	38	ffvelrnda 6623	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
40		fof 6366	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
41	40	adantr 474	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐴⟶𝐵)
42	41	ffvelrnda 6623	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
43		resieq 5657	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
44	39, 42, 43	syl2anc 579	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
45	44	adantlr 705	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
46	36, 45	mpbid 224	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
47	46	eqcomd 2783	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
48	8, 10, 47	eqfnfvd 6577	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
49	48	ex 403	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
50	6, 49	impbid 204	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2106  ⟨cop 4403   class class class wbr 4886   I cid 5260  ^◡ccnv 5354   ↾ cres 5357   ∘ ccom 5359   Fn wfn 6130  ⟶wf 6131  –onto→wfo 6133  ‘cfv 6135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143

This theorem is referenced by: (None)