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Mirrors > Home > MPE Home > Th. List > elhoma | 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) |
homaval.j | ⊢ 𝐽 = (Hom ‘𝐶) |
homaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
homaval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
elhoma | ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homarcl.h | . . . 4 ⊢ 𝐻 = (Homa‘𝐶) | |
2 | homafval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | homafval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | homaval.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐶) | |
5 | homaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | homaval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | homaval 16681 | . . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))) |
8 | 7 | breqd 4664 | . 2 ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ 𝑍({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))𝐹)) |
9 | brxp 5147 | . . 3 ⊢ (𝑍({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {〈𝑋, 𝑌〉} ∧ 𝐹 ∈ (𝑋𝐽𝑌))) | |
10 | opex 4932 | . . . . 5 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
11 | 10 | elsn2 4211 | . . . 4 ⊢ (𝑍 ∈ {〈𝑋, 𝑌〉} ↔ 𝑍 = 〈𝑋, 𝑌〉) |
12 | 11 | anbi1i 731 | . . 3 ⊢ ((𝑍 ∈ {〈𝑋, 𝑌〉} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
13 | 9, 12 | bitri 264 | . 2 ⊢ (𝑍({〈𝑋, 𝑌〉} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
14 | 8, 13 | syl6bb 276 | 1 ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 〈cop 4183 class class class wbr 4653 × cxp 5112 ‘cfv 5888 (class class class)co 6650 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-ov 6653 df-homa 16676 |
This theorem is referenced by: elhomai 16683 homa1 16687 homahom2 16688 |
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