Theorem fcof1od 7307

Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7300 and fcofo 7301. Formerly part of proof of fcof1o 7309. (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 7300	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)
4	1, 2, 3	syl2anc 582	. 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 7301	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)
8	1, 5, 6, 7	syl3anc 1368	. 2 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
9		df-f1o 6558	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
10	4, 8, 9	sylanbrc 581	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   = wceq 1533   I cid 5577   ↾ cres 5682   ∘ ccom 5684  ⟶wf 6547  –1-1→wf1 6548  –onto→wfo 6549  –1-1-onto→wf1o 6550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559

This theorem is referenced by:  2fcoidinvd  7308  fcof1o  7309  2fvidf1od  7311  catciso  18105  pmtrff1o  19423  evpmodpmf1o  21533