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Mirrors > Home > MPE Home > Th. List > homahom2 | Structured version Visualization version GIF version |
Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
homahom.j | ⊢ 𝐽 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
homahom2 | ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5067 | . . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ 〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌)) | |
2 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 2 | homarcl 17288 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
5 | homahom.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐶) | |
6 | 2, 3 | homarcl2 17295 | . . . . . 6 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
7 | 6 | simpld 497 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶)) |
8 | 6 | simprd 498 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶)) |
9 | 2, 3, 4, 5, 7, 8 | elhoma 17292 | . . . 4 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
10 | 1, 9 | sylbi 219 | . . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
11 | 10 | ibi 269 | . 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
12 | 11 | simprd 498 | 1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 Homachoma 17283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-homa 17286 |
This theorem is referenced by: homahom 17299 |
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