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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 6717 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | homafval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | 5 | sqxpeqd 5583 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵)) |
7 | fveq2 6717 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
8 | homafval.j | . . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶) | |
9 | 7, 8 | eqtr4di 2796 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽) |
10 | 9 | fveq1d 6719 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥)) |
11 | 10 | xpeq2d 5581 | . . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥))) |
12 | 6, 11 | mpteq12dv 5140 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
13 | df-homa 17532 | . . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 4 | fvexi 6731 | . . . . . 6 ⊢ 𝐵 ∈ V |
15 | 14, 14 | xpex 7538 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
16 | 15 | mptex 7039 | . . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V |
17 | 12, 13, 16 | fvmpt 6818 | . . 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 2790 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4541 ↦ cmpt 5135 × cxp 5549 ‘cfv 6380 Basecbs 16760 Hom chom 16813 Catccat 17167 Homachoma 17529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-homa 17532 |
This theorem is referenced by: homaf 17536 homaval 17537 |
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