Theorem 2fvcoidd 7333

Description: Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first 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)

Assertion

Ref	Expression
2fvcoidd	⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))

Distinct variable groups:   𝐴,𝑎   𝐹,𝑎   𝐺,𝑎

Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)

Proof of Theorem 2fvcoidd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fvcoidd.g	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
2		2fvcoidd.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3		fcompt 7167	. . 3 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
4	1, 2, 3	syl2anc 583	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
5		2fvcoidd.i	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)
6		2fveq3 6925	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
7		id 22	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → 𝑎 = 𝑥)
8	6, 7	eqeq12d 2756	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝐺‘(𝐹‘𝑎)) = 𝑎 ↔ (𝐺‘(𝐹‘𝑥)) = 𝑥))
9	8	rspccv 3632	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎 → (𝑥 ∈ 𝐴 → (𝐺‘(𝐹‘𝑥)) = 𝑥))
10	5, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐺‘(𝐹‘𝑥)) = 𝑥))
11	10	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(𝐹‘𝑥)) = 𝑥)
12	11	mpteq2dva 5266	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝑥))
13		mptresid 6080	. . 3 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
14	12, 13	eqtr4di 2798	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = ( I ↾ 𝐴))
15	4, 14	eqtrd 2780	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249   I cid 5592   ↾ cres 5702   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  2fvidf1od  7334  2fvidinvd  7335