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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 17635 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
6 | oveq12 7411 | . . 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 7437 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V) | |
11 | 5, 7, 8, 9, 10 | ovmpod 7553 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 Basecbs 17145 Hom chom 17209 Homf chomf 17611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-homf 17615 |
This theorem is referenced by: homfeqval 17642 comfffval2 17646 comffval2 17647 comfval2 17648 catsubcat 17790 subcss2 17794 fullsubc 17801 fullresc 17802 funcres2c 17855 hof1 18211 hofcllem 18215 hofcl 18216 yonffthlem 18239 srhmsubc 20568 srhmsubcALTV 47213 |
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