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Theorem fcof1od 7043
 Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7036 and fcofo 7037. Formerly part of proof of fcof1o 7045. (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 7036 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐺𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
41, 2, 3syl2anc 588 . 2 (𝜑𝐹:𝐴1-1𝐵)
5 fcof1od.g . . 3 (𝜑𝐺:𝐵𝐴)
6 fcof1od.b . . 3 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
7 fcofo 7037 . . 3 ((𝐹:𝐴𝐵𝐺:𝐵𝐴 ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
81, 5, 6, 7syl3anc 1369 . 2 (𝜑𝐹:𝐴onto𝐵)
9 df-f1o 6343 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
104, 8, 9sylanbrc 587 1 (𝜑𝐹:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   I cid 5430   ↾ cres 5527   ∘ ccom 5529  ⟶wf 6332  –1-1→wf1 6333  –onto→wfo 6334  –1-1-onto→wf1o 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344 This theorem is referenced by:  2fcoidinvd  7044  fcof1o  7045  2fvidf1od  7047  catciso  17434  pmtrff1o  18659  evpmodpmf1o  20362
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