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Theorem 2fvidf1od 6518
 Description: A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
2fvcoidd.f (𝜑𝐹:𝐴𝐵)
2fvcoidd.g (𝜑𝐺:𝐵𝐴)
2fvcoidd.i (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)
2fvidf1od.i (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)
Assertion
Ref Expression
2fvidf1od (𝜑𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐴,𝑎   𝐹,𝑎   𝐺,𝑎   𝐵,𝑏   𝐹,𝑏   𝐺,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑏)   𝐵(𝑎)

Proof of Theorem 2fvidf1od
StepHypRef Expression
1 2fvcoidd.f . 2 (𝜑𝐹:𝐴𝐵)
2 2fvcoidd.g . 2 (𝜑𝐺:𝐵𝐴)
3 2fvcoidd.i . . 3 (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)
41, 2, 32fvcoidd 6517 . 2 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
5 2fvidf1od.i . . 3 (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)
62, 1, 52fvcoidd 6517 . 2 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
71, 2, 4, 6fcof1od 6514 1 (𝜑𝐹:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480  ∀wral 2908  ⟶wf 5853  –1-1-onto→wf1o 5856  ‘cfv 5857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865 This theorem is referenced by:  m2cpminv  20505  foresf1o  29231
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