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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 5359 | . . 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 2823 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
7 | 3, 4, 5, 6 | idafval 17319 | . 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ((Id‘𝐶)‘𝑥)〉)) |
8 | idaf.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
9 | eqid 2823 | . . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
10 | 8, 9 | homarw 17308 | . . 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 17322 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥)) |
14 | 10, 13 | sseldi 3967 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐴) |
15 | 2, 7, 14 | fmpt2d 6889 | 1 ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cotp 4577 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Catccat 16937 Idccid 16938 Arrowcarw 17284 Homachoma 17285 Idacida 17315 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-cat 16941 df-cid 16942 df-homa 17288 df-arw 17289 df-ida 17317 |
This theorem is referenced by: (None) |
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