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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 5476 | . . 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 2735 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
7 | 3, 4, 5, 6 | idafval 18111 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉)) |
8 | idaf.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
9 | eqid 2735 | . . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
10 | 8, 9 | homarw 18100 | . . 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 18114 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥)) |
14 | 10, 13 | sselid 3993 | . 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 1537 ∈ wcel 2106 Vcvv 3478 〈cotp 4639 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Catccat 17709 Idccid 17710 Arrowcarw 18076 Homachoma 18077 Idacida 18107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-cat 17713 df-cid 17714 df-homa 18080 df-arw 18081 df-ida 18109 |
This theorem is referenced by: (None) |
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