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Mirrors > Home > MPE Home > Th. List > elhomai | Structured version Visualization version GIF version |
Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homarcl.h | ⊢ 𝐻 = (Homa‘𝐶) |
homafval.b | ⊢ 𝐵 = (Base‘𝐶) |
homafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
homaval.j | ⊢ 𝐽 = (Hom ‘𝐶) |
homaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
homaval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
elhomai.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌)) |
Ref | Expression |
---|---|
elhomai | ⊢ (𝜑 → 〈𝑋, 𝑌〉(𝑋𝐻𝑌)𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2735 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉) | |
2 | elhomai.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌)) | |
3 | homarcl.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | homafval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
5 | homafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | homaval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐶) | |
7 | homaval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | homaval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 3, 4, 5, 6, 7, 8 | elhoma 18085 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉(𝑋𝐻𝑌)𝐹 ↔ (〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
10 | 1, 2, 9 | mpbir2and 713 | 1 ⊢ (𝜑 → 〈𝑋, 𝑌〉(𝑋𝐻𝑌)𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 〈cop 4636 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Hom chom 17308 Catccat 17708 Homachoma 18076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-homa 18079 |
This theorem is referenced by: elhomai2 18087 |
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