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Theorem 2fvidf1od 7243
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 7242 . 2 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
5 2fvidf1od.i . . 3 (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)
62, 1, 52fvcoidd 7242 . 2 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
71, 2, 4, 6fcof1od 7239 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wral 3064  wf 6492  1-1-ontowf1o 6495  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504
This theorem is referenced by:  m2cpminv  22107  foresf1o  31429  sticksstones11  40555  sticksstones12  40557  sticksstones19  40564  metakunt14  40581
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