f1eqcocnv - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > f1eqcocnv		GIF version

Theorem f1eqcocnv 5987

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

**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 5664	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
2		coeq2 4933	. . . . 5 ⊢ (𝐹 = 𝐺 → (^◡𝐹 ∘ 𝐹) = (^◡𝐹 ∘ 𝐺))
3	2	eqeq1d 2247	. . . 4 ⊢ (𝐹 = 𝐺 → ((^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴) ↔ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
4	1, 3	syl5ibcom 155	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 = 𝐺 → (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
5	4	adantr 276	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 → (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))
6		f1fn 5595	. . . . . . 7 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺 Fn 𝐴)
7	6	adantl 277	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐺 Fn 𝐴)
8	7	adantr 276	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴)
9		f1fn 5595	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
10	9	adantr 276	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
11	10	adantr 276	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴)
12		equid 1753	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
13		resieq 5068	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥( I ↾ 𝐴)𝑥 ↔ 𝑥 = 𝑥))
14	12, 13	mpbiri 168	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
15	14	anidms 401	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥( I ↾ 𝐴)𝑥)
16	15	adantl 277	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥)
17		breq 4127	. . . . . . . 8 ⊢ ((^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → (𝑥(^◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥))
18	17	ad2antlr 493	. . . . . . 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 167	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥(^◡𝐹 ∘ 𝐺)𝑥)
20		vex 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
21	20, 20	brco 4946	. . . . . . . . 9 ⊢ (𝑥(^◡𝐹 ∘ 𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦 ∧ 𝑦^◡𝐹𝑥))
22		fnfun 5473	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
23	7, 22	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐺)
24	23	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → Fun 𝐺)
25		fndm 5475	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
26	7, 25	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐺 = 𝐴)
27	26	eleq2d 2308	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝐴))
28	27	biimpar 297	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐺)
29		funopfvb 5738	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ dom 𝐺) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
30	24, 28, 29	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
31	30	bicomd 141	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝐺‘𝑥) = 𝑦))
32		df-br 4126	. . . . . . . . . . . . 13 ⊢ (𝑥𝐺𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)
33		eqcom 2240	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = 𝑦)
34	31, 32, 33	3bitr4g 223	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 ↔ 𝑦 = (𝐺‘𝑥)))
35	34	biimpd 144	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 → 𝑦 = (𝐺‘𝑥)))
36		df-br 4126	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
37		fnfun 5473	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
38	10, 37	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐹)
39	38	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → Fun 𝐹)
40		fndm 5475	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
41	10, 40	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴)
42	41	eleq2d 2308	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
43	42	biimpar 297	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
44		funopfvb 5738	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
45	39, 43, 44	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
46	36, 45	bitr4id 199	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 ↔ (𝐹‘𝑥) = 𝑦))
47		vex 2824	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
48	47, 20	brcnv 4958	. . . . . . . . . . . . 13 ⊢ (𝑦^◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
49		eqcom 2240	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
50	46, 48, 49	3bitr4g 223	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦^◡𝐹𝑥 ↔ 𝑦 = (𝐹‘𝑥)))
51	50	biimpd 144	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦^◡𝐹𝑥 → 𝑦 = (𝐹‘𝑥)))
52	35, 51	anim12d 335	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝐺𝑦 ∧ 𝑦^◡𝐹𝑥) → (𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
53	52	eximdv 1933	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑥𝐺𝑦 ∧ 𝑦^◡𝐹𝑥) → ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
54	21, 53	biimtrid 152	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(^◡𝐹 ∘ 𝐺)𝑥 → ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
55	6	anim1i 340	. . . . . . . . . 10 ⊢ ((𝐺:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
56	55	adantll 480	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
57		funfvex 5707	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ V)
58	57	funfni 5478	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ V)
59		eqvincg 2950	. . . . . . . . 9 ⊢ ((𝐺‘𝑥) ∈ V → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
60	56, 58, 59	3syl 17	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))))
61	54, 60	sylibrd 169	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(^◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥)))
62	61	adantlr 481	. . . . . 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)))
63	19, 62	mpd 13	. . . . 5 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
64	8, 11, 63	eqfnfvd 5800	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹)
65	64	eqcomd 2244	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺)
66	65	ex 115	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → ((^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺))
67	5, 66	impbid 129	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (^◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∃wex 1545 ∈ wcel 2209 Vcvv 2821 ⟨cop 3708 class class class wbr 4125 I cid 4428 ^◡ccnv 4768 dom cdm 4769 ↾ cres 4771 ∘ ccom 4773 Fun wfun 5366 Fn wfn 5367 –1-1→wf1 5369 ‘cfv 5372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380

This theorem is referenced by: (None)

W3C validator