Theorem foeqcnvco 7328

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⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → (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 6871	. . . 4 ⊢ (𝐹:𝐴⟶onto𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))
2		cnveq 5857	. . . . . 6 ⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺)
3	2	coeq2d 5845	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐹) = (𝐹 ∘ ◡𝐺))
4	3	eqeq1d 2765	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵) ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
5	1, 4	syl5ibcom 248	. . 3 ⊢ (𝐹:𝐴⟶onto𝐵 → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
6	5	adantr 486	. 2 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))
7		fofn 6817	. . . . 5 ⊢ (𝐹:𝐴⟶onto𝐵 → 𝐹 Fn 𝐴)
8	7	ad2antrr 739	. . . 4 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9		fofn 6817	. . . . 5 ⊢ (𝐺:𝐴⟶onto𝐵 → 𝐺 Fn 𝐴)
10	9	ad2antlr 740	. . . 4 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
11	9	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐺 Fn 𝐴)
12		fnopfv 7096	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
13	11, 12	sylan 592	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐺‘𝑥)⟩ ∈ 𝐺)
14		fvex 6918	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑥) ∈ V
15		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5866	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑥)◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥))
17		df-br 5107	. . . . . . . . . . . 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⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)◡𝐺𝑥)
20	7	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐹 Fn 𝐴)
21		fnopfv 7096	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
22	20, 21	sylan 592	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
23		df-br 5107	. . . . . . . . . . 11 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
24	22, 23	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
25		breq2 5110	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐺‘𝑥)◡𝐺𝑦 ↔ (𝐺‘𝑥)◡𝐺𝑥))
26		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦𝐹(𝐹‘𝑥) ↔ 𝑥𝐹(𝐹‘𝑥)))
27	25, 26	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)) ↔ ((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥))))
28	15, 27	spcev 3564	. . . . . . . . . 10 ⊢ (((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
29	19, 24, 28	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
30		fvex 6918	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) ∈ V
31	14, 30	brco 5854	. . . . . . . . 9 ⊢ ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))
32	29, 31	sylibr 237	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
33	32	adantlr 728	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥))
34		breq 5108	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)))
35	34	ad2antlr 740	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (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⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))
37		fof 6815	. . . . . . . . . 10 ⊢ (𝐺:𝐴⟶onto𝐵 → 𝐺:𝐴⟶𝐵)
38	37	adantl 487	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐺:𝐴⟶𝐵)
39	38	ffvelcdmda 7105	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
40		fof 6815	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶onto𝐵 → 𝐹:𝐴⟶𝐵)
41	40	adantr 486	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → 𝐹:𝐴⟶𝐵)
42	41	ffvelcdmda 7105	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
43		resieq 5991	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
44	39, 42, 43	syl2anc 596	. . . . . . 7 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
45	44	adantlr 728	. . . . . 6 ⊢ ((((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)))
46	36, 45	mpbid 235	. . . . 5 ⊢ ((((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
47	46	eqcomd 2769	. . . 4 ⊢ ((((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
48	8, 10, 47	eqfnfvd 7052	. . 3 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
49	48	ex 418	. 2 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
50	6, 49	impbid 215	1 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐴⟶onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ⟨cop 4838   class class class wbr 5106   I cid 5549  ^◡ccnv 5654   ↾ cres 5657   ∘ ccom 5659   Fn wfn 6550  ⟶wf 6551  ⟶ontowfo 6553  ‘cfv 6555

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-fv 6563

This theorem is used by: (None)