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Mirrors > Home > MPE Home > Th. List > homarw | Structured version Visualization version GIF version |
Description: A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwhoma.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homarw | ⊢ (𝑋𝐻𝑌) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovssunirn 7194 | . 2 ⊢ (𝑋𝐻𝑌) ⊆ ∪ ran 𝐻 | |
2 | arwrcl.a | . . 3 ⊢ 𝐴 = (Arrow‘𝐶) | |
3 | arwhoma.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | arwval 17305 | . 2 ⊢ 𝐴 = ∪ ran 𝐻 |
5 | 1, 4 | sseqtrri 4006 | 1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3938 ∪ cuni 4840 ran crn 5558 ‘cfv 6357 (class class class)co 7158 Arrowcarw 17284 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-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-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 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-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-homa 17288 df-arw 17289 |
This theorem is referenced by: idaf 17325 homdmcoa 17329 coaval 17330 coapm 17333 |
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