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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 16952 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
6 | oveq12 7144 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
7 | 6 | adantl 485 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
8 | homfval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | homfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | ovexd 7170 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V) | |
11 | 5, 7, 8, 9, 10 | ovmpod 7281 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Basecbs 16475 Hom chom 16568 Homf chomf 16929 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-homf 16933 |
This theorem is referenced by: homfeqval 16959 comfffval2 16963 comffval2 16964 comfval2 16965 catsubcat 17101 subcss2 17105 fullsubc 17112 fullresc 17113 funcres2c 17163 hof1 17496 hofcllem 17500 hofcl 17501 yonffthlem 17524 srhmsubc 44700 srhmsubcALTV 44718 |
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