Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6191 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | homafval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | 5 | sqxpeqd 5141 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵)) |
7 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
8 | homafval.j | . . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶) | |
9 | 7, 8 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽) |
10 | 9 | fveq1d 6193 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥)) |
11 | 10 | xpeq2d 5139 | . . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥))) |
12 | 6, 11 | mpteq12dv 4733 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
13 | df-homa 16676 | . . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝐶) ∈ V | |
15 | 4, 14 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
16 | 15, 15 | xpex 6962 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
17 | 16 | mptex 6486 | . . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V |
18 | 12, 13, 17 | fvmpt 6282 | . . 3 ⊢ (𝐶 ∈ Cat → (Homa‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
19 | 2, 18 | syl 17 | . 2 ⊢ (𝜑 → (Homa‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
20 | 1, 19 | syl5eq 2668 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 Basecbs 15857 Hom chom 15952 Catccat 16325 Homachoma 16673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-homa 16676 |
This theorem is referenced by: homaf 16680 homaval 16681 |
Copyright terms: Public domain | W3C validator |