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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 2822 | . 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 17286 | . 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 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Catccat 16929 Homachoma 17277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-homa 17280 |
This theorem is referenced by: elhomai2 17288 |
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