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Mirrors > Home > MPE Home > Th. List > idaf | Structured version Visualization version GIF version |
Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
idafval.i | ⊢ 𝐼 = (Ida‘𝐶) |
idafval.b | ⊢ 𝐵 = (Base‘𝐶) |
idafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
idaf.a | ⊢ 𝐴 = (Arrow‘𝐶) |
Ref | Expression |
---|---|
idaf | ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | otex 5082 | . . 3 ⊢ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉 ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉 ∈ V) |
3 | idafval.i | . . 3 ⊢ 𝐼 = (Ida‘𝐶) | |
4 | idafval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
5 | idafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | eqid 2760 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
7 | 3, 4, 5, 6 | idafval 16908 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉)) |
8 | idaf.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
9 | eqid 2760 | . . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
10 | 8, 9 | homarw 16897 | . . 3 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ 𝐴 |
11 | 5 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat) |
12 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
13 | 3, 4, 11, 12, 9 | idahom 16911 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥)) |
14 | 10, 13 | sseldi 3742 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐴) |
15 | 2, 7, 14 | fmpt2d 6556 | 1 ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 〈cotp 4329 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 Catccat 16526 Idccid 16527 Arrowcarw 16873 Homachoma 16874 Idacida 16904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-ot 4330 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-cat 16530 df-cid 16531 df-homa 16877 df-arw 16878 df-ida 16906 |
This theorem is referenced by: (None) |
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