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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 17298 | . . . 4 ⊢ Rel (𝑋𝐻𝑌) |
3 | 1st2nd 7740 | . . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
5 | 1st2ndbr 7743 | . . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
6 | 2, 5 | mpan 688 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
7 | 1 | homa1 17299 | . . . . 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 4811 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 〈(1st ‘𝐹), (2nd ‘𝐹)〉 = 〈〈𝑋, 𝑌〉, (2nd ‘𝐹)〉) |
10 | 4, 9 | eqtrd 2858 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈〈𝑋, 𝑌〉, (2nd ‘𝐹)〉) |
11 | df-ot 4578 | . 2 ⊢ 〈𝑋, 𝑌, (2nd ‘𝐹)〉 = 〈〈𝑋, 𝑌〉, (2nd ‘𝐹)〉 | |
12 | 10, 11 | syl6eqr 2876 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 〈cotp 4577 class class class wbr 5068 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 Homachoma 17285 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-homa 17288 |
This theorem is referenced by: arwdmcd 17314 arwlid 17334 arwrid 17335 |
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