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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 7127 | . . 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 6890 | . . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥))) | |
7 | id 22 | . . . . . . . 8 ⊢ (𝑎 = 𝑥 → 𝑎 = 𝑥) | |
8 | 6, 7 | eqeq12d 2742 | . . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝐺‘(𝐹‘𝑎)) = 𝑎 ↔ (𝐺‘(𝐹‘𝑥)) = 𝑥)) |
9 | 8 | rspccv 3603 | . . . . . 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 5241 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝑥)) |
13 | mptresid 6044 | . . 3 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥) | |
14 | 12, 13 | eqtr4di 2784 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))) = ( I ↾ 𝐴)) |
15 | 4, 14 | eqtrd 2766 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ↦ cmpt 5224 I cid 5566 ↾ cres 5671 ∘ ccom 5673 ⟶wf 6533 ‘cfv 6537 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 |
This theorem is referenced by: 2fvidf1od 7292 2fvidinvd 7293 |
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