Theorem 2fvcoidd 7275

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 7109	. . 3 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
4	1, 2, 3	syl2anc 593	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
5		2fvcoidd.i	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎)
6		2fveq3 6866	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
7		id 22	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → 𝑎 = 𝑥)
8	6, 7	eqeq12d 2777	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝐺‘(𝐹‘𝑎)) = 𝑎 ↔ (𝐺‘(𝐹‘𝑥)) = 𝑥))
9	8	rspccv 3577	. . . . . 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 410	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(𝐹‘𝑥)) = 𝑥)
12	11	mpteq2dva 5190	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝑥))
13		mptresid 6035	. . 3 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
14	12, 13	eqtr4di 2814	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = ( I ↾ 𝐴))
15	4, 14	eqtrd 2796	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ↦ cmpt 5178   I cid 5537   ↾ cres 5645   ∘ ccom 5647  ⟶wf 6511  ‘cfv 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523

This theorem is referenced by:  2fvidf1od  7276  2fvidinvd  7277