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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 17380 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
6 | oveq12 7277 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
7 | 6 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
8 | homfval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | homfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | ovexd 7303 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V) | |
11 | 5, 7, 8, 9, 10 | ovmpod 7416 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ‘cfv 6430 (class class class)co 7268 ∈ cmpo 7270 Basecbs 16893 Hom chom 16954 Homf chomf 17356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-homf 17360 |
This theorem is referenced by: homfeqval 17387 comfffval2 17391 comffval2 17392 comfval2 17393 catsubcat 17535 subcss2 17539 fullsubc 17546 fullresc 17547 funcres2c 17598 hof1 17953 hofcllem 17957 hofcl 17958 yonffthlem 17981 srhmsubc 45586 srhmsubcALTV 45604 |
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