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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 18022 | . . . 4 ⊢ Rel (𝑋𝐻𝑌) |
3 | 1st2nd 8039 | . . . 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 8042 | . . . . . 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 18023 | . . . . 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 4873 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd ‘𝐹)⟩) |
10 | 4, 9 | eqtrd 2765 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2nd ‘𝐹)⟩) |
11 | df-ot 4631 | . 2 ⊢ ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd ‘𝐹)⟩ | |
12 | 10, 11 | eqtr4di 2783 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟨cop 4628 ⟨cotp 4630 class class class wbr 5141 Rel wrel 5675 ‘cfv 6541 (class class class)co 7414 1st c1st 7987 2nd c2nd 7988 Homachoma 18009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-ot 4631 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-1st 7989 df-2nd 7990 df-homa 18012 |
This theorem is referenced by: arwdmcd 18038 arwlid 18058 arwrid 18059 |
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