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| Mirrors > Home > MPE Home > Th. List > fvssunirn | Structured version Visualization version GIF version | ||
| Description: The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by SN, 13-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| fvssunirn | ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elfvunirn 6937 | . 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹) | |
| 2 | 1 | ssriv 3986 | 1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3950 ∪ cuni 4906 ran crn 5685 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: ovssunirn 7468 marypha2lem1 9476 acnlem 10089 fin23lem29 10382 itunitc 10462 hsmexlem5 10471 wunfv 10773 wunex2 10779 strfvss 17225 prdsvallem 17500 prdsval 17501 prdsbas 17503 prdsplusg 17504 prdsmulr 17505 prdsvsca 17506 prdshom 17513 mreunirn 17645 mrcfval 17652 mrcssv 17658 mrisval 17674 sscpwex 17860 wunfunc 17947 catcxpccl 18253 comppfsc 23541 filunirn 23891 elflim 23980 flffval 23998 fclsval 24017 isfcls 24018 fcfval 24042 tsmsxplem1 24162 xmetunirn 24348 mopnval 24449 tmsval 24494 cfilfval 25299 caufval 25310 issgon 34125 elrnsiga 34128 volmeas 34233 omssubadd 34303 neibastop2lem 36362 ctbssinf 37408 ismtyval 37808 dicval 41179 prjcrv0 42648 ismrc 42717 nacsfix 42728 hbt 43147 | 
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