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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 5106 | . . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ 〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌)) | |
2 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 2 | homarcl 17913 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
5 | homahom.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐶) | |
6 | 2, 3 | homarcl2 17920 | . . . . . 6 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
7 | 6 | simpld 495 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶)) |
8 | 6 | simprd 496 | . . . . 5 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶)) |
9 | 2, 3, 4, 5, 7, 8 | elhoma 17917 | . . . 4 ⊢ (〈𝑍, 𝐹〉 ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
10 | 1, 9 | sylbi 216 | . . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌)))) |
11 | 10 | ibi 266 | . 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = 〈𝑋, 𝑌〉 ∧ 𝐹 ∈ (𝑋𝐽𝑌))) |
12 | 11 | simprd 496 | 1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4592 class class class wbr 5105 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 Hom chom 17143 Homachoma 17908 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-homa 17911 |
This theorem is referenced by: homahom 17924 |
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