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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 16703 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
6 | oveq12 6915 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
7 | 6 | adantl 475 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
8 | homfval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | homfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | ovexd 6940 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V) | |
11 | 5, 7, 8, 9, 10 | ovmpt2d 7049 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3415 ‘cfv 6124 (class class class)co 6906 ↦ cmpt2 6908 Basecbs 16223 Hom chom 16317 Homf chomf 16680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-homf 16684 |
This theorem is referenced by: homfeqval 16710 comfffval2 16714 comffval2 16715 comfval2 16716 catsubcat 16852 subcss2 16856 fullsubc 16863 fullresc 16864 funcres2c 16914 hof1 17248 hofcllem 17252 hofcl 17253 yonffthlem 17276 srhmsubc 42924 srhmsubcALTV 42942 |
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