Theorem foeqcnvco 7302

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 6851	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))
2		cnveq 5863	. . . . . 6 ⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺)
3	2	coeq2d 5852	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐹) = (𝐹 ∘ ◡𝐺))
4	3	eqeq1d 2772	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵) ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
5	1, 4	syl5ibcom 248	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
6	5	adantr 485	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
7		fofn 6798	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
8	7	ad2antrr 738	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9		fofn 6798	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺 Fn 𝐴)
10	9	ad2antlr 739	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
11	9	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺 Fn 𝐴)
12		fnopfv 7074	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
13	11, 12	sylan 591	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
14		fvex 6898	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑥) ∈ V
15		vex 3466	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5872	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥))
17		df-br 5115	. . . . . . . . . . . 12 ⊢ (𝑥𝐺(𝐺‘𝑥) ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
18	16, 17	bitri 278	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
19	13, 18	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)◡𝐺𝑥)
20	7	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹 Fn 𝐴)
21		fnopfv 7074	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
22	20, 21	sylan 591	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
23		df-br 5115	. . . . . . . . . . 11 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
24	22, 23	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
25		breq2 5118	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐺‘𝑥)◡𝐺𝑦 ↔ (𝐺‘𝑥)◡𝐺𝑥))
26		breq1 5117	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦𝐹(𝐹‘𝑥) ↔ 𝑥𝐹(𝐹‘𝑥)))
27	25, 26	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)) ↔ ((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥))))
28	15, 27	spcev 3573	. . . . . . . . . 10 ⊢ (((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
29	19, 24, 28	syl2anc 595	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
30		fvex 6898	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
31	14, 30	brco 5860	. . . . . . . . 9 ⊢ ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
32	29, 31	sylibr 237	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
33	32	adantlr 727	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
34		breq 5116	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)))
35	34	ad2antlr 739	. . . . . . 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)))
36	33, 35	mpbid 235	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))
37		fof 6796	. . . . . . . . . 10 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
38	37	adantl 486	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴⟶𝐵)
39	38	ffvelcdmda 7083	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
40		fof 6796	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
41	40	adantr 485	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐴⟶𝐵)
42	41	ffvelcdmda 7083	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
43		resieq 5993	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
44	39, 42, 43	syl2anc 595	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
45	44	adantlr 727	. . . . . 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)))
46	36, 45	mpbid 235	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
47	46	eqcomd 2776	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
48	8, 10, 47	eqfnfvd 7032	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
49	48	ex 417	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
50	6, 49	impbid 215	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2150  ⟨cop 4600   class class class wbr 5114   I cid 5559  ^◡ccnv 5664   ↾ cres 5667   ∘ ccom 5669   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548

This theorem is referenced by: (None)