Theorem f1eqcocnv 7299

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 6851	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
2		coeq2 5844	. . . . 5 ⊢ (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ 𝐺))
3	2	eqeq1d 2765	. . . 4 ⊢ (𝐹 = 𝐺 → ((◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴) ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
4	1, 3	syl5ibcom 248	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
5	4	adantr 485	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
6		f1fn 6775	. . . . . . 7 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺 Fn 𝐴)
7	6	adantl 486	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐺 Fn 𝐴)
8	7	adantr 485	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴)
9		f1fn 6775	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
10	9	adantr 485	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
11	10	adantr 485	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴)
12		equid 2042	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
13		resieq 5989	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥( I ↾ 𝐴)𝑥 ↔ 𝑥 = 𝑥))
14	12, 13	mpbiri 261	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
15	14	anidms 576	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥( I ↾ 𝐴)𝑥)
16	15	adantl 486	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
17		breq 5111	. . . . . . . 8 ⊢ ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥))
18	17	ad2antlr 739	. . . . . . 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 260	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥(◡𝐹 ∘ 𝐺)𝑥)
20		fnfun 6635	. . . . . . . . . . . . . . 15 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
21	7, 20	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐺)
22	7	fndmd 6640	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐺 = 𝐴)
23	22	eleq2d 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝐴))
24	23	biimpar 482	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐺)
25		funopfvb 6935	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ dom 𝐺) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26	21, 24, 25	syl2an2r 697	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
27	26	bicomd 226	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝐺‘𝑥) = 𝑦))
28		df-br 5110	. . . . . . . . . . . 12 ⊢ (𝑥𝐺𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)
29		eqcom 2770	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = 𝑦)
30	27, 28, 29	3bitr4g 317	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 ↔ 𝑦 = (𝐺‘𝑥)))
31	30	biimpd 232	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 → 𝑦 = (𝐺‘𝑥)))
32		df-br 5110	. . . . . . . . . . . . 13 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
33		fnfun 6635	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
34	10, 33	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐹)
35	10	fndmd 6640	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴)
36	35	eleq2d 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
37	36	biimpar 482	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
38		funopfvb 6935	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
39	34, 37, 38	syl2an2r 697	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
40	32, 39	bitr4id 293	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 ↔ (𝐹‘𝑥) = 𝑦))
41		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
42		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43	41, 42	brcnv 5868	. . . . . . . . . . . 12 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
44		eqcom 2770	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
45	40, 43, 44	3bitr4g 317	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 ↔ 𝑦 = (𝐹‘𝑥)))
46	45	biimpd 232	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 → 𝑦 = (𝐹‘𝑥)))
47	31, 46	anim12d 620	. . . . . . . . 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 1947	. . . . . . . 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 5856	. . . . . . . 8 ⊢ (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥))
50		fvex 6894	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
51	50	eqvinc 3608	. . . . . . . 8 ⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥)))
52	48, 49, 51	3imtr4g 299	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥)))
53	52	adantlr 727	. . . . . 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 16	. . . . 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 7028	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹)
56	55	eqcomd 2769	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺)
57	56	ex 417	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺))
58	5, 57	impbid 215	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109   I cid 5555  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663   ∘ ccom 5665  Fun wfun 6530   Fn wfn 6531  –1-1→wf1 6533  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  weisoeq  7353