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Mirrors > Home > MPE Home > Th. List > homadmcd | Structured version Visualization version GIF version |
Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homadmcd | ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homahom.h | . . . . 5 ⊢ 𝐻 = (Homa‘𝐶) | |
2 | 1 | homarel 16733 | . . . 4 ⊢ Rel (𝑋𝐻𝑌) |
3 | 1st2nd 7258 | . . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) | |
4 | 2, 3 | mpan 706 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
5 | 1st2ndbr 7261 | . . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
6 | 2, 5 | mpan 706 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
7 | 1 | homa1 16734 | . . . . 5 ⊢ ((1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹) → (1st ‘𝐹) = 〈𝑋, 𝑌〉) |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹) = 〈𝑋, 𝑌〉) |
9 | 8 | opeq1d 4439 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈(1st ‘𝐹), (2nd ‘𝐹)〉 = 〈〈𝑋, 𝑌〉, (2nd ‘𝐹)〉) |
10 | 4, 9 | eqtrd 2685 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈〈𝑋, 𝑌〉, (2nd ‘𝐹)〉) |
11 | df-ot 4219 | . 2 ⊢ 〈𝑋, 𝑌, (2nd ‘𝐹)〉 = 〈〈𝑋, 𝑌〉, (2nd ‘𝐹)〉 | |
12 | 10, 11 | syl6eqr 2703 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 〈cop 4216 〈cotp 4218 class class class wbr 4685 Rel wrel 5148 ‘cfv 5926 (class class class)co 6690 1st c1st 7208 2nd c2nd 7209 Homachoma 16720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-ot 4219 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-1st 7210 df-2nd 7211 df-homa 16723 |
This theorem is referenced by: arwdmcd 16749 arwlid 16769 arwrid 16770 |
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