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Mirrors > Home > MPE Home > Th. List > homdmcoa | Structured version Visualization version GIF version |
Description: If 𝐹:𝑋⟶𝑌 and 𝐺:𝑌⟶𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homdmcoa.o | ⊢ · = (compa‘𝐶) |
homdmcoa.h | ⊢ 𝐻 = (Homa‘𝐶) |
homdmcoa.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
homdmcoa.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) |
Ref | Expression |
---|---|
homdmcoa | ⊢ (𝜑 → 𝐺dom · 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶) | |
2 | homdmcoa.h | . . . 4 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | 1, 2 | homarw 17298 | . . 3 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶) |
4 | homdmcoa.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
5 | 3, 4 | sseldi 3913 | . 2 ⊢ (𝜑 → 𝐹 ∈ (Arrow‘𝐶)) |
6 | 1, 2 | homarw 17298 | . . 3 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶) |
7 | homdmcoa.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
8 | 6, 7 | sseldi 3913 | . 2 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶)) |
9 | 2 | homacd 17293 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝜑 → (coda‘𝐹) = 𝑌) |
11 | 2 | homadm 17292 | . . . 4 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌) |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝜑 → (doma‘𝐺) = 𝑌) |
13 | 10, 12 | eqtr4d 2836 | . 2 ⊢ (𝜑 → (coda‘𝐹) = (doma‘𝐺)) |
14 | homdmcoa.o | . . 3 ⊢ · = (compa‘𝐶) | |
15 | 14, 1 | eldmcoa 17317 | . 2 ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ (Arrow‘𝐶) ∧ 𝐺 ∈ (Arrow‘𝐶) ∧ (coda‘𝐹) = (doma‘𝐺))) |
16 | 5, 8, 13, 15 | syl3anbrc 1340 | 1 ⊢ (𝜑 → 𝐺dom · 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 domacdoma 17272 codaccoda 17273 Arrowcarw 17274 Homachoma 17275 compaccoa 17306 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-doma 17276 df-coda 17277 df-homa 17278 df-arw 17279 df-coa 17308 |
This theorem is referenced by: (None) |
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