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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 5357 | . . 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 2821 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
7 | 3, 4, 5, 6 | idafval 17317 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉)) |
8 | idaf.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
9 | eqid 2821 | . . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
10 | 8, 9 | homarw 17306 | . . 3 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ 𝐴 |
11 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat) |
12 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
13 | 3, 4, 11, 12, 9 | idahom 17320 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥)) |
14 | 10, 13 | sseldi 3965 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐴) |
15 | 2, 7, 14 | fmpt2d 6887 | 1 ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cotp 4575 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Catccat 16935 Idccid 16936 Arrowcarw 17282 Homachoma 17283 Idacida 17313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-cat 16939 df-cid 16940 df-homa 17286 df-arw 17287 df-ida 17315 |
This theorem is referenced by: (None) |
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