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Theorem fcof1od 6712
Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 6705 and fcofo 6706. Formerly part of proof of fcof1o 6714. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
fcof1od.f (𝜑𝐹:𝐴𝐵)
fcof1od.g (𝜑𝐺:𝐵𝐴)
fcof1od.a (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
fcof1od.b (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
Assertion
Ref Expression
fcof1od (𝜑𝐹:𝐴1-1-onto𝐵)

Proof of Theorem fcof1od
StepHypRef Expression
1 fcof1od.f . . 3 (𝜑𝐹:𝐴𝐵)
2 fcof1od.a . . 3 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
3 fcof1 6705 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐺𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
41, 2, 3syl2anc 696 . 2 (𝜑𝐹:𝐴1-1𝐵)
5 fcof1od.g . . 3 (𝜑𝐺:𝐵𝐴)
6 fcof1od.b . . 3 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
7 fcofo 6706 . . 3 ((𝐹:𝐴𝐵𝐺:𝐵𝐴 ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
81, 5, 6, 7syl3anc 1477 . 2 (𝜑𝐹:𝐴onto𝐵)
9 df-f1o 6056 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
104, 8, 9sylanbrc 701 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632   I cid 5173  cres 5268  ccom 5270  wf 6045  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  2fcoidinvd  6713  fcof1o  6714  2fvidf1od  6716  catciso  16958  pmtrff1o  18083  evpmodpmf1o  20144
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