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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 5470 | . . 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 2737 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
| 7 | 3, 4, 5, 6 | idafval 18102 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉)) | 
| 8 | idaf.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
| 9 | eqid 2737 | . . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
| 10 | 8, 9 | homarw 18091 | . . 3 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ 𝐴 | 
| 11 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat) | 
| 12 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 13 | 3, 4, 11, 12, 9 | idahom 18105 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥)) | 
| 14 | 10, 13 | sselid 3981 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐴) | 
| 15 | 2, 7, 14 | fmpt2d 7144 | 1 ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cotp 4634 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Catccat 17707 Idccid 17708 Arrowcarw 18067 Homachoma 18068 Idacida 18098 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-cat 17711 df-cid 17712 df-homa 18071 df-arw 18072 df-ida 18100 | 
| This theorem is referenced by: (None) | 
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