Theorem 2fvidinvd 7245

Description: Show that two functions are inverse to each other by applying them twice to each value of their domains. (Contributed by AV, 13-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
2fvidinvd	⊢ (𝜑 → ◡𝐹 = 𝐺)

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 2fvidinvd

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 7243	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
5		2fvidf1od.i	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏)
6	2, 1, 5	2fvcoidd 7243	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
7	1, 2, 4, 6	2fcoidinvd 7241	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1541  ∀wral 3064  ^◡ccnv 5632  ⟶wf 6492  ‘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  22109  metakunt14  40590