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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 2737 | . 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 17910 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉(𝑋𝐻𝑌)𝐹 ↔ (〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
10 | 1, 2, 9 | mpbir2and 711 | 1 ⊢ (𝜑 → 〈𝑋, 𝑌〉(𝑋𝐻𝑌)𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 〈cop 4590 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 Hom chom 17136 Catccat 17536 Homachoma 17901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-homa 17904 |
This theorem is referenced by: elhomai2 17912 |
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