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Mirrors > Home > MPE Home > Th. List > homfval | Structured version Visualization version GIF version |
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
homffval.f | ⊢ 𝐹 = (Homf ‘𝐶) |
homffval.b | ⊢ 𝐵 = (Base‘𝐶) |
homffval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
homfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
homfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
homfval | ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homffval.f | . . . 4 ⊢ 𝐹 = (Homf ‘𝐶) | |
2 | homffval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | homffval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | homffval 16959 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
6 | oveq12 7164 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
7 | 6 | adantl 484 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
8 | homfval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | homfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | ovexd 7190 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V) | |
11 | 5, 7, 8, 9, 10 | ovmpod 7301 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 Basecbs 16482 Hom chom 16575 Homf chomf 16936 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-homf 16940 |
This theorem is referenced by: homfeqval 16966 comfffval2 16970 comffval2 16971 comfval2 16972 catsubcat 17108 subcss2 17112 fullsubc 17119 fullresc 17120 funcres2c 17170 hof1 17503 hofcllem 17507 hofcl 17508 yonffthlem 17531 srhmsubc 44346 srhmsubcALTV 44364 |
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