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Mirrors > Home > MPE Home > Th. List > homaval | Structured version Visualization version GIF version |
Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homarcl.h | ⊢ 𝐻 = (Homa‘𝐶) |
homafval.b | ⊢ 𝐵 = (Base‘𝐶) |
homafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
homaval.j | ⊢ 𝐽 = (Hom ‘𝐶) |
homaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
homaval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
homaval | ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7138 | . 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
2 | homarcl.h | . . . 4 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | homafval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | homafval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | homaval.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐶) | |
6 | 2, 3, 4, 5 | homafval 17281 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧)))) |
7 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
8 | 7 | sneqd 4537 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → {𝑧} = {〈𝑋, 𝑌〉}) |
9 | 7 | fveq2d 6649 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝐽‘〈𝑋, 𝑌〉)) |
10 | df-ov 7138 | . . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘〈𝑋, 𝑌〉) | |
11 | 9, 10 | eqtr4di 2851 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝑋𝐽𝑌)) |
12 | 8, 11 | xpeq12d 5550 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → ({𝑧} × (𝐽‘𝑧)) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
13 | homaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
14 | homaval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
15 | 13, 14 | opelxpd 5557 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
16 | snex 5297 | . . . . 5 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
17 | ovex 7168 | . . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V | |
18 | 16, 17 | xpex 7456 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V) |
20 | 6, 12, 15, 19 | fvmptd 6752 | . 2 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
21 | 1, 20 | syl5eq 2845 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 × cxp 5517 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 Catccat 16927 Homachoma 17275 |
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-homa 17278 |
This theorem is referenced by: elhoma 17284 |
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