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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 7153 | . 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 17283 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧)))) |
7 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
8 | 7 | sneqd 4573 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → {𝑧} = {〈𝑋, 𝑌〉}) |
9 | 7 | fveq2d 6669 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝐽‘〈𝑋, 𝑌〉)) |
10 | df-ov 7153 | . . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘〈𝑋, 𝑌〉) | |
11 | 9, 10 | syl6eqr 2874 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝑋𝐽𝑌)) |
12 | 8, 11 | xpeq12d 5581 | . . 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 5588 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
16 | snex 5324 | . . . . 5 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
17 | ovex 7183 | . . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V | |
18 | 16, 17 | xpex 7470 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V) |
20 | 6, 12, 15, 19 | fvmptd 6770 | . 2 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
21 | 1, 20 | syl5eq 2868 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 {csn 4561 〈cop 4567 × cxp 5548 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Catccat 16929 Homachoma 17277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-homa 17280 |
This theorem is referenced by: elhoma 17286 |
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