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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 17998 | . . . 4 ⊢ Rel (𝑋𝐻𝑌) |
3 | 1st2nd 8024 | . . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) | |
4 | 2, 3 | mpan 687 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) |
5 | 1st2ndbr 8027 | . . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
6 | 2, 5 | mpan 687 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
7 | 1 | homa1 17999 | . . . . 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 4874 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd ‘𝐹)⟩) |
10 | 4, 9 | eqtrd 2766 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2nd ‘𝐹)⟩) |
11 | df-ot 4632 | . 2 ⊢ ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd ‘𝐹)⟩ | |
12 | 10, 11 | eqtr4di 2784 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 ⟨cotp 4631 class class class wbr 5141 Rel wrel 5674 ‘cfv 6537 (class class class)co 7405 1st c1st 7972 2nd c2nd 7973 Homachoma 17985 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-1st 7974 df-2nd 7975 df-homa 17988 |
This theorem is referenced by: arwdmcd 18014 arwlid 18034 arwrid 18035 |
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