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Theorem fcof1od 7047
Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7040 and fcofo 7041. Formerly part of proof of fcof1o 7049. (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 7040 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐺𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
41, 2, 3syl2anc 584 . 2 (𝜑𝐹:𝐴1-1𝐵)
5 fcof1od.g . . 3 (𝜑𝐺:𝐵𝐴)
6 fcof1od.b . . 3 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
7 fcofo 7041 . . 3 ((𝐹:𝐴𝐵𝐺:𝐵𝐴 ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
81, 5, 6, 7syl3anc 1365 . 2 (𝜑𝐹:𝐴onto𝐵)
9 df-f1o 6358 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
104, 8, 9sylanbrc 583 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530   I cid 5457  cres 5555  ccom 5557  wf 6347  1-1wf1 6348  ontowfo 6349  1-1-ontowf1o 6350
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359
This theorem is referenced by:  2fcoidinvd  7048  fcof1o  7049  2fvidf1od  7051  catciso  17359  pmtrff1o  18513  evpmodpmf1o  20656
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