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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 6906 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
| 4 | homafval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
| 6 | 5 | sqxpeqd 5717 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵)) |
| 7 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
| 8 | homafval.j | . . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽) |
| 10 | 9 | fveq1d 6908 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥)) |
| 11 | 10 | xpeq2d 5715 | . . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥))) |
| 12 | 6, 11 | mpteq12dv 5233 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
| 13 | df-homa 18071 | . . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
| 14 | 4 | fvexi 6920 | . . . . . 6 ⊢ 𝐵 ∈ V |
| 15 | 14, 14 | xpex 7773 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
| 16 | 15 | mptex 7243 | . . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V |
| 17 | 12, 13, 16 | fvmpt 7016 | . . 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 | eqtrid 2789 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4626 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 Basecbs 17247 Hom chom 17308 Catccat 17707 Homachoma 18068 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-homa 18071 |
| This theorem is referenced by: homaf 18075 homaval 18076 |
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