Theorem f1eqcocnv 7246

Description: Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Proof shortened by Wolf Lammen, 29-May-2024.)

Assertion

Ref	Expression
f1eqcocnv	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))

Proof of Theorem f1eqcocnv

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

Step	Hyp	Ref	Expression
1		f1cocnv1 6798	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
2		coeq2 5801	. . . . 5 ⊢ (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ 𝐺))
3	2	eqeq1d 2741	. . . 4 ⊢ (𝐹 = 𝐺 → ((◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴) ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
4	1, 3	syl5ibcom 246	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
5	4	adantr 481	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
6		f1fn 6725	. . . . . . 7 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺 Fn 𝐴)
7	6	adantl 482	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐺 Fn 𝐴)
8	7	adantr 481	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴)
9		f1fn 6725	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
10	9	adantr 481	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
11	10	adantr 481	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴)
12		equid 2019	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
13		resieq 5943	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥( I ↾ 𝐴)𝑥 ↔ 𝑥 = 𝑥))
14	12, 13	mpbiri 259	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
15	14	anidms 571	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥( I ↾ 𝐴)𝑥)
16	15	adantl 482	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
17		breq 5075	. . . . . . . 8 ⊢ ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥))
18	17	ad2antlr 733	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥))
19	16, 18	mpbird 258	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥(◡𝐹 ∘ 𝐺)𝑥)
20		fnfun 6586	. . . . . . . . . . . . . . 15 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
21	7, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐺)
22	7	fndmd 6591	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐺 = 𝐴)
23	22	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝐴))
24	23	biimpar 478	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐺)
25		funopfvb 6882	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ dom 𝐺) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26	21, 24, 25	syl2an2r 691	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
27	26	bicomd 224	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝐺‘𝑥) = 𝑦))
28		df-br 5074	. . . . . . . . . . . 12 ⊢ (𝑥𝐺𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)
29		eqcom 2746	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = 𝑦)
30	27, 28, 29	3bitr4g 315	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 ↔ 𝑦 = (𝐺‘𝑥)))
31	30	biimpd 230	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 → 𝑦 = (𝐺‘𝑥)))
32		df-br 5074	. . . . . . . . . . . . 13 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
33		fnfun 6586	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
34	10, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐹)
35	10	fndmd 6591	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴)
36	35	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
37	36	biimpar 478	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
38		funopfvb 6882	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
39	34, 37, 38	syl2an2r 691	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
40	32, 39	bitr4id 291	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 ↔ (𝐹‘𝑥) = 𝑦))
41		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
42		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43	41, 42	brcnv 5825	. . . . . . . . . . . 12 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
44		eqcom 2746	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
45	40, 43, 44	3bitr4g 315	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 ↔ 𝑦 = (𝐹‘𝑥)))
46	45	biimpd 230	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 → 𝑦 = (𝐹‘𝑥)))
47	31, 46	anim12d 615	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥) → (𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
48	47	eximdv 1924	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥) → ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
49	42, 42	brco 5813	. . . . . . . 8 ⊢ (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥))
50		fvex 6841	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
51	50	eqvinc 3587	. . . . . . . 8 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥)))
52	48, 49, 51	3imtr4g 297	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥)))
53	52	adantlr 721	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥)))
54	19, 53	mpd 15	. . . . 5 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
55	8, 11, 54	eqfnfvd 6975	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹)
56	55	eqcomd 2745	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺)
57	56	ex 413	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺))
58	5, 57	impbid 213	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ⟨cop 4562   class class class wbr 5073   I cid 5513  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   ∘ ccom 5623  Fun wfun 6480   Fn wfn 6481  –1-1→wf1 6483  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by:  weisoeq  7300