Theorem foeqcnvco 7050

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 6634	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))
2		cnveq 5738	. . . . . 6 ⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺)
3	2	coeq2d 5727	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐹) = (𝐹 ∘ ◡𝐺))
4	3	eqeq1d 2823	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵) ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
5	1, 4	syl5ibcom 247	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
6	5	adantr 483	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
7		fofn 6586	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
8	7	ad2antrr 724	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9		fofn 6586	. . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺 Fn 𝐴)
10	9	ad2antlr 725	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
11	9	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺 Fn 𝐴)
12		fnopfv 6837	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
13	11, 12	sylan 582	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
14		fvex 6677	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑥) ∈ V
15		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5747	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥))
17		df-br 5059	. . . . . . . . . . . 12 ⊢ (𝑥𝐺(𝐺‘𝑥) ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
18	16, 17	bitri 277	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
19	13, 18	sylibr 236	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)◡𝐺𝑥)
20	7	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹 Fn 𝐴)
21		fnopfv 6837	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
22	20, 21	sylan 582	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
23		df-br 5059	. . . . . . . . . . 11 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
24	22, 23	sylibr 236	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
25		breq2 5062	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐺‘𝑥)◡𝐺𝑦 ↔ (𝐺‘𝑥)◡𝐺𝑥))
26		breq1 5061	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦𝐹(𝐹‘𝑥) ↔ 𝑥𝐹(𝐹‘𝑥)))
27	25, 26	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)) ↔ ((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥))))
28	15, 27	spcev 3606	. . . . . . . . . 10 ⊢ (((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
29	19, 24, 28	syl2anc 586	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
30		fvex 6677	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
31	14, 30	brco 5735	. . . . . . . . 9 ⊢ ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
32	29, 31	sylibr 236	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
33	32	adantlr 713	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
34		breq 5060	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)))
35	34	ad2antlr 725	. . . . . . 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 234	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))
37		fof 6584	. . . . . . . . . 10 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵)
38	37	adantl 484	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴⟶𝐵)
39	38	ffvelrnda 6845	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
40		fof 6584	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
41	40	adantr 483	. . . . . . . . 9 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐴⟶𝐵)
42	41	ffvelrnda 6845	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
43		resieq 5858	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
44	39, 42, 43	syl2anc 586	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
45	44	adantlr 713	. . . . . 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 234	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
47	46	eqcomd 2827	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
48	8, 10, 47	eqfnfvd 6799	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
49	48	ex 415	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
50	6, 49	impbid 214	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ⟨cop 4566   class class class wbr 5058   I cid 5453  ^◡ccnv 5548   ↾ cres 5551   ∘ ccom 5553   Fn wfn 6344  ⟶wf 6345  –onto→wfo 6347  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357

This theorem is referenced by: (None)