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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 5485 | . . 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 2740 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
7 | 3, 4, 5, 6 | idafval 18124 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉)) |
8 | idaf.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
9 | eqid 2740 | . . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
10 | 8, 9 | homarw 18113 | . . 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 18127 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥)) |
14 | 10, 13 | sselid 4006 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐴) |
15 | 2, 7, 14 | fmpt2d 7158 | 1 ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 Vcvv 3488 〈cotp 4656 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 Catccat 17722 Idccid 17723 Arrowcarw 18089 Homachoma 18090 Idacida 18120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-cat 17726 df-cid 17727 df-homa 18093 df-arw 18094 df-ida 18122 |
This theorem is referenced by: (None) |
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