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Theorem 2fvidf1od 6453
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 6452 . 2 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
5 2fvidf1od.i . . 3 (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)
62, 1, 52fvcoidd 6452 . 2 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
71, 2, 4, 6fcof1od 6449 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wral 2896  wf 5800  1-1-ontowf1o 5803  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812
This theorem is referenced by:  m2cpminv  20384  foresf1o  28727
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