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Mirrors > Home > MPE Home > Th. List > 2fvcoidd | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
2fvcoidd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
2fvcoidd.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
2fvcoidd.i | ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) |
Ref | Expression |
---|---|
2fvcoidd | ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fvcoidd.g | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
2 | 2fvcoidd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | fcompt 7130 | . . 3 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥)))) | |
4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥)))) |
5 | 2fvcoidd.i | . . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) | |
6 | 2fveq3 6896 | . . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥))) | |
7 | id 22 | . . . . . . . 8 ⊢ (𝑎 = 𝑥 → 𝑎 = 𝑥) | |
8 | 6, 7 | eqeq12d 2748 | . . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝐺‘(𝐹‘𝑎)) = 𝑎 ↔ (𝐺‘(𝐹‘𝑥)) = 𝑥)) |
9 | 8 | rspccv 3609 | . . . . . 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 407 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(𝐹‘𝑥)) = 𝑥) |
12 | 11 | mpteq2dva 5248 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝑥)) |
13 | mptresid 6050 | . . 3 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥) | |
14 | 12, 13 | eqtr4di 2790 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = ( I ↾ 𝐴)) |
15 | 4, 14 | eqtrd 2772 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ↦ cmpt 5231 I cid 5573 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 |
This theorem is referenced by: 2fvidf1od 7295 2fvidinvd 7296 |
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