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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 7162 | . 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 17292 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧)))) |
7 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
8 | 7 | sneqd 4582 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → {𝑧} = {〈𝑋, 𝑌〉}) |
9 | 7 | fveq2d 6677 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝐽‘〈𝑋, 𝑌〉)) |
10 | df-ov 7162 | . . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘〈𝑋, 𝑌〉) | |
11 | 9, 10 | syl6eqr 2877 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐽‘𝑧) = (𝑋𝐽𝑌)) |
12 | 8, 11 | xpeq12d 5589 | . . 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 5596 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
16 | snex 5335 | . . . . 5 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
17 | ovex 7192 | . . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V | |
18 | 16, 17 | xpex 7479 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌)) ∈ V) |
20 | 6, 12, 15, 19 | fvmptd 6778 | . 2 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
21 | 1, 20 | syl5eq 2871 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 〈cop 4576 × cxp 5556 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Hom chom 16579 Catccat 16938 Homachoma 17286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-homa 17289 |
This theorem is referenced by: elhoma 17295 |
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