Theorem 2fvidf1od 7276

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 7275	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
5		2fvidf1od.i	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏)
6	2, 1, 5	2fvcoidd 7275	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
7	1, 2, 4, 6	fcof1od 7272	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   = wceq 1540  ∀wral 3045  ⟶wf 6510  –1-1-onto→wf1o 6513  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by:  m2cpminv  22654  foresf1o  32440  primrootscoprbij  42097  sticksstones11  42151  sticksstones12  42153  sticksstones19  42160