Theorem 2fvidf1od 7318

Description: A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
2fvcoidd.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2fvcoidd.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
2fvcoidd.i	⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)
2fvidf1od.i	⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏)

Assertion

Ref	Expression
2fvidf1od	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

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

Step	Hyp	Ref	Expression
1		2fvcoidd.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		2fvcoidd.g	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
3		2fvcoidd.i	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)
4	1, 2, 3	2fvcoidd 7317	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
5		2fvidf1od.i	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏)
6	2, 1, 5	2fvcoidd 7317	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
7	1, 2, 4, 6	fcof1od 7314	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   = wceq 1540  ∀wral 3061  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569

This theorem is referenced by:  m2cpminv  22766  foresf1o  32523  primrootscoprbij  42103  sticksstones11  42157  sticksstones12  42159  sticksstones19  42166  metakunt14  42219