Theorem fcof1od 7303

Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7296 and fcofo 7297. Formerly part of proof of fcof1o 7305. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
fcof1od.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcof1od.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
fcof1od.a	⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
fcof1od.b	⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))

Assertion

Ref	Expression
fcof1od	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Proof of Theorem fcof1od

Step	Hyp	Ref	Expression
1		fcof1od.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		fcof1od.a	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
3		fcof1 7296	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)
4	1, 2, 3	syl2anc 596	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
5		fcof1od.g	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
6		fcof1od.b	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
7		fcofo 7297	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)
8	1, 5, 6, 7	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
9		df-f1o 6546	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
10	4, 8, 9	sylanbrc 595	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   I cid 5544   ↾ cres 5652   ∘ ccom 5654  ⟶wf 6535  –1-1→wf1 6536  –onto→wfo 6537  –1-1-onto→wf1o 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547

This theorem is used by:  2fcoidinvd  7304  fcof1o  7305  2fvidf1od  7307  catciso  18292  pmtrff1o  19683  evpmodpmf1o  21908