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Theorem 2fvidf1od 6825
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 6824 . 2 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
5 2fvidf1od.i . . 3 (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)
62, 1, 52fvcoidd 6824 . 2 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
71, 2, 4, 6fcof1od 6821 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wral 3090  wf 6131  1-1-ontowf1o 6134  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143
This theorem is referenced by:  m2cpminv  20972  foresf1o  29905
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