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| Mirrors > Home > MPE Home > Th. List > ovssunirn | Structured version Visualization version GIF version | ||
| Description: The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| ovssunirn | ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ov 7434 | . 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
| 2 | fvssunirn 6939 | . 2 ⊢ (𝐹‘〈𝑋, 𝑌〉) ⊆ ∪ ran 𝐹 | |
| 3 | 1, 2 | eqsstri 4030 | 1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3951 〈cop 4632 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 (class class class)co 7431 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: prdsvallem 17499 prdsplusg 17503 prdsmulr 17504 prdsvsca 17505 prdshom 17512 wunfunc 17946 wunnat 18004 homarw 18091 catcoppccl 18162 catcfuccl 18163 catcxpccl 18252 isanmbfmOLD 34256 isanmbfm 34258 | 
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