Theorem 2fvcoidd 7312

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 7148	. . 3 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
4	1, 2, 3	syl2anc 582	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
5		2fvcoidd.i	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)
6		2fveq3 6907	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
7		id 22	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → 𝑎 = 𝑥)
8	6, 7	eqeq12d 2744	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝐺‘(𝐹‘𝑎)) = 𝑎 ↔ (𝐺‘(𝐹‘𝑥)) = 𝑥))
9	8	rspccv 3608	. . . . . 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 405	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(𝐹‘𝑥)) = 𝑥)
12	11	mpteq2dva 5252	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝑥))
13		mptresid 6059	. . 3 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
14	12, 13	eqtr4di 2786	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = ( I ↾ 𝐴))
15	4, 14	eqtrd 2768	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3058   ↦ cmpt 5235   I cid 5579   ↾ cres 5684   ∘ ccom 5686  ⟶wf 6549  ‘cfv 6553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561

This theorem is referenced by:  2fvidf1od  7313  2fvidinvd  7314