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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 7438 | . 2 ⊢ (𝑋𝐻𝑌) ⊆ ∪ ran 𝐻 | |
2 | arwrcl.a | . . 3 ⊢ 𝐴 = (Arrow‘𝐶) | |
3 | arwhoma.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | arwval 17997 | . 2 ⊢ 𝐴 = ∪ ran 𝐻 |
5 | 1, 4 | sseqtrri 4012 | 1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊆ wss 3941 ∪ cuni 4900 ran crn 5668 ‘cfv 6534 (class class class)co 7402 Arrowcarw 17976 Homachoma 17977 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-homa 17980 df-arw 17981 |
This theorem is referenced by: idaf 18017 homdmcoa 18021 coaval 18022 coapm 18025 |
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