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Mirrors > Home > MPE Home > Th. List > homa1 | Structured version Visualization version GIF version |
Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homa1 | ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝑍 = 〈𝑋, 𝑌〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5031 | . . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ 〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌)) | |
2 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 2 | homarcl 17280 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
5 | eqid 2798 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | 2, 3 | homarcl2 17287 | . . . . . 6 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
7 | 6 | simpld 498 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶)) |
8 | 6 | simprd 499 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶)) |
9 | 2, 3, 4, 5, 7, 8 | elhoma 17284 | . . . 4 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))) |
10 | 1, 9 | sylbi 220 | . . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))) |
11 | 10 | ibi 270 | . 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))) |
12 | 11 | simpld 498 | 1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝑍 = 〈𝑋, 𝑌〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 Homachoma 17275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-homa 17278 |
This theorem is referenced by: homadm 17292 homacd 17293 homadmcd 17294 |
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