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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 6497 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | homafval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2827 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | 5 | sqxpeqd 5436 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵)) |
7 | fveq2 6497 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
8 | homafval.j | . . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶) | |
9 | 7, 8 | syl6eqr 2827 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽) |
10 | 9 | fveq1d 6499 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥)) |
11 | 10 | xpeq2d 5434 | . . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥))) |
12 | 6, 11 | mpteq12dv 5009 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
13 | df-homa 17157 | . . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 4 | fvexi 6511 | . . . . . 6 ⊢ 𝐵 ∈ V |
15 | 14, 14 | xpex 7292 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
16 | 15 | mptex 6811 | . . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V |
17 | 12, 13, 16 | fvmpt 6594 | . . 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 2821 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {csn 4436 ↦ cmpt 5005 × cxp 5402 ‘cfv 6186 Basecbs 16338 Hom chom 16431 Catccat 16806 Homachoma 17154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-homa 17157 |
This theorem is referenced by: homaf 17161 homaval 17162 |
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