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Mirrors > Home > MPE Home > Th. List > homafval | 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) |
homafval.j | ⊢ 𝐽 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
homafval | ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homarcl.h | . 2 ⊢ 𝐻 = (Homa‘𝐶) | |
2 | homafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | fveq2 6663 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | homafval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | 5 | sqxpeqd 5580 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵)) |
7 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
8 | homafval.j | . . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶) | |
9 | 7, 8 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽) |
10 | 9 | fveq1d 6665 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥)) |
11 | 10 | xpeq2d 5578 | . . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥))) |
12 | 6, 11 | mpteq12dv 5142 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
13 | df-homa 17274 | . . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 4 | fvexi 6677 | . . . . . 6 ⊢ 𝐵 ∈ V |
15 | 14, 14 | xpex 7465 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
16 | 15 | mptex 6977 | . . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V |
17 | 12, 13, 16 | fvmpt 6761 | . . 3 ⊢ (𝐶 ∈ Cat → (Homa‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝜑 → (Homa‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
19 | 1, 18 | syl5eq 2865 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 ↦ cmpt 5137 × cxp 5546 ‘cfv 6348 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-homa 17274 |
This theorem is referenced by: homaf 17278 homaval 17279 |
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