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Mirrors > Home > MPE Home > Th. List > elhoma | 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 |
---|---|
elhoma | ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homarcl.h | . . . 4 ⊢ 𝐻 = (Homa‘𝐶) | |
2 | homafval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | homafval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | homaval.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐶) | |
5 | homaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | homaval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | homaval 17279 | . . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
8 | 7 | breqd 5068 | . 2 ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ 𝑍({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))𝐹)) |
9 | brxp 5594 | . . 3 ⊢ (𝑍({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {〈𝑋, 𝑌〉} ∧ 𝐹 ∈ (𝑋𝐽𝑌))) | |
10 | opex 5347 | . . . . 5 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
11 | 10 | elsn2 4594 | . . . 4 ⊢ (𝑍 ∈ {〈𝑋, 𝑌〉} ↔ 𝑍 = 〈𝑋, 𝑌〉) |
12 | 11 | anbi1i 623 | . . 3 ⊢ ((𝑍 ∈ {〈𝑋, 𝑌〉} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
13 | 9, 12 | bitri 276 | . 2 ⊢ (𝑍({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
14 | 8, 13 | syl6bb 288 | 1 ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 class class class wbr 5057 × cxp 5546 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Hom chom 16564 Catccat 16923 Homachoma 17271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-homa 17274 |
This theorem is referenced by: elhomai 17281 homa1 17285 homahom2 17286 |
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