foeqcnvco - Intuitionistic Logic Explorer

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Theorem foeqcnvco 5837

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 5530	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ^◡𝐹) = ( I ↾ 𝐵))
2		cnveq 4840	. . . . . 6 ⊢ (𝐹 = 𝐺 → ^◡𝐹 = ^◡𝐺)
3	2	coeq2d 4828	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ ^◡𝐹) = (𝐹 ∘ ^◡𝐺))
4	3	eqeq1d 2205	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹 ∘ ^◡𝐹) = ( I ↾ 𝐵) ↔ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)))
5	1, 4	syl5ibcom 155	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 = 𝐺 → (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)))
6	5	adantr 276	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 → (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)))
7		fofn 5482	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
8	7	ad2antrr 488	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9		fofn 5482	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺 Fn 𝐴)
10	9	ad2antlr 489	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
11	9	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺 Fn 𝐴)
12		fnopfv 5692	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
13	11, 12	sylan 283	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
14	9	anim1i 340	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
15	14	adantll 476	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
16		funfvex 5575	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ V)
17	16	funfni 5358	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ V)
18		vex 2766	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
19		brcnvg 4847	. . . . . . . . . . . . . 14 ⊢ (((𝐺‘𝑥) ∈ V ∧ 𝑥 ∈ V) → ((𝐺‘𝑥)^◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥)))
20	17, 18, 19	sylancl 413	. . . . . . . . . . . . 13 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)^◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥)))
21		df-br 4034	. . . . . . . . . . . . 13 ⊢ (𝑥𝐺(𝐺‘𝑥) ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
22	20, 21	bitrdi 196	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)^◡𝐺𝑥 ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺))
23	15, 22	syl 14	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)^◡𝐺𝑥 ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺))
24	13, 23	mpbird 167	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)^◡𝐺𝑥)
25	7	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹 Fn 𝐴)
26		fnopfv 5692	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
27	25, 26	sylan 283	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
28		df-br 4034	. . . . . . . . . . 11 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
29	27, 28	sylibr 134	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
30		breq2 4037	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐺‘𝑥)^◡𝐺𝑦 ↔ (𝐺‘𝑥)^◡𝐺𝑥))
31		breq1 4036	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦𝐹(𝐹‘𝑥) ↔ 𝑥𝐹(𝐹‘𝑥)))
32	30, 31	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (((𝐺‘𝑥)^◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)) ↔ ((𝐺‘𝑥)^◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥))))
33	18, 32	spcev 2859	. . . . . . . . . 10 ⊢ (((𝐺‘𝑥)^◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)) → ∃𝑦((𝐺‘𝑥)^◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
34	24, 29, 33	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑦((𝐺‘𝑥)^◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
35	15, 17	syl 14	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ V)
36	7	anim1i 340	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
37	36	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
38		funfvex 5575	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
39	38	funfni 5358	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
40	37, 39	syl 14	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
41		brcog 4833	. . . . . . . . . 10 ⊢ (((𝐺‘𝑥) ∈ V ∧ (𝐹‘𝑥) ∈ V) → ((𝐺‘𝑥)(𝐹 ∘ ^◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)^◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥))))
42	35, 40, 41	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)(𝐹 ∘ ^◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)^◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥))))
43	34, 42	mpbird 167	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ^◡𝐺)(𝐹‘𝑥))
44	43	adantlr 477	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ^◡𝐺)(𝐹‘𝑥))
45		breq 4035	. . . . . . . 8 ⊢ ((𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵) → ((𝐺‘𝑥)(𝐹 ∘ ^◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)))
46	45	ad2antlr 489	. . . . . . 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)))
47	44, 46	mpbid 147	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))
48		fof 5480	. . . . . . . . . 10 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
49	48	adantl 277	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴⟶𝐵)
50	49	ffvelcdmda 5697	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
51		fof 5480	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
52	51	adantr 276	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐴⟶𝐵)
53	52	ffvelcdmda 5697	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
54		resieq 4956	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
55	50, 53, 54	syl2anc 411	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
56	55	adantlr 477	. . . . . 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)))
57	47, 56	mpbid 147	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
58	57	eqcomd 2202	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
59	8, 10, 58	eqfnfvd 5662	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
60	59	ex 115	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
61	6, 60	impbid 129	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ^◡𝐺) = ( I ↾ 𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1506 ∈ wcel 2167 Vcvv 2763 ⟨cop 3625 class class class wbr 4033 I cid 4323 ^◡ccnv 4662 ↾ cres 4665 ∘ ccom 4667 Fn wfn 5253 ⟶wf 5254 –onto→wfo 5256 ‘cfv 5258

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-sbc 2990 df-csb 3085 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fo 5264 df-fv 5266

This theorem is referenced by: (None)

W3C validator