Theorem f1eqcocnv 7308

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 6855	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
2		coeq2 5846	. . . . 5 ⊢ (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ 𝐺))
3	2	eqeq1d 2767	. . . 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 486	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
6		f1fn 6779	. . . . . . 7 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺 Fn 𝐴)
7	6	adantl 487	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐺 Fn 𝐴)
8	7	adantr 486	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴)
9		f1fn 6779	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
10	9	adantr 486	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
11	10	adantr 486	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴)
12		equid 2045	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
13		resieq 5991	. . . . . . . . . 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 577	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥( I ↾ 𝐴)𝑥)
16	15	adantl 487	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
17		breq 5113	. . . . . . . 8 ⊢ ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥))
18	17	ad2antlr 740	. . . . . . 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 6639	. . . . . . . . . . . . . . 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 6644	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐺 = 𝐴)
23	22	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝐴))
24	23	biimpar 483	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐺)
25		funopfvb 6939	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ dom 𝐺) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26	21, 24, 25	syl2an2r 698	. . . . . . . . . . . . 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 5112	. . . . . . . . . . . 12 ⊢ (𝑥𝐺𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)
29		eqcom 2772	. . . . . . . . . . . 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 5112	. . . . . . . . . . . . 13 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
33		fnfun 6639	. . . . . . . . . . . . . . 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 6644	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴)
36	35	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
37	36	biimpar 483	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
38		funopfvb 6939	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
39	34, 37, 38	syl2an2r 698	. . . . . . . . . . . . 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 3461	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
42		vex 3461	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
43	41, 42	brcnv 5870	. . . . . . . . . . . 12 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
44		eqcom 2772	. . . . . . . . . . . 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 621	. . . . . . . . 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 1950	. . . . . . . 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 5858	. . . . . . . 8 ⊢ (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥))
50		fvex 6898	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
51	50	eqvinc 3610	. . . . . . . 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 728	. . . . . 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 7032	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹)
56	55	eqcomd 2771	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺)
57	56	ex 418	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ⟨cop 4597   class class class wbr 5111   I cid 5557  ^◡ccnv 5662  dom cdm 5663   ↾ cres 5665   ∘ ccom 5667  Fun wfun 6534   Fn wfn 6535  –1-1→wf1 6537  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is used by:  weisoeq  7364