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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 6552 | . 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 16502 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧)))) |
7 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
8 | 7 | sneqd 4137 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → {𝑧} = {〈𝑋, 𝑌〉}) |
9 | 7 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝐽‘〈𝑋, 𝑌〉)) |
10 | df-ov 6552 | . . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘〈𝑋, 𝑌〉) | |
11 | 9, 10 | syl6eqr 2662 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝑋𝐽𝑌)) |
12 | 8, 11 | xpeq12d 5064 | . . 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 | opelxpi 5072 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
16 | 13, 14, 15 | syl2anc 691 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
17 | snex 4835 | . . . . 5 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
18 | ovex 6577 | . . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V | |
19 | 17, 18 | xpex 6860 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V |
20 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V) |
21 | 6, 12, 16, 20 | fvmptd 6197 | . 2 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
22 | 1, 21 | syl5eq 2656 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 × cxp 5036 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 Catccat 16148 Homachoma 16496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-homa 16499 |
This theorem is referenced by: elhoma 16505 |
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