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Theorem 2fvidf1od 7245
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 7244 . 2 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
5 2fvidf1od.i . . 3 (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)
62, 1, 52fvcoidd 7244 . 2 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
71, 2, 4, 6fcof1od 7241 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wral 3061  wf 6493  1-1-ontowf1o 6496  cfv 6497
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  m2cpminv  22125  foresf1o  31474  sticksstones11  40610  sticksstones12  40612  sticksstones19  40619  metakunt14  40636
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