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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 6924 | . 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹) | |
2 | 1 | ssriv 3987 | 1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3949 ∪ cuni 4909 ran crn 5678 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-cnv 5685 df-dm 5687 df-rn 5688 df-iota 6496 df-fv 6552 |
This theorem is referenced by: ovssunirn 7445 marypha2lem1 9430 acnlem 10043 fin23lem29 10336 itunitc 10416 hsmexlem5 10425 wunfv 10727 wunex2 10733 strfvss 17120 prdsvallem 17400 prdsval 17401 prdsbas 17403 prdsplusg 17404 prdsmulr 17405 prdsvsca 17406 prdshom 17413 mreunirn 17545 mrcfval 17552 mrcssv 17558 mrisval 17574 sscpwex 17762 wunfunc 17849 wunfuncOLD 17850 catcxpccl 18159 catcxpcclOLD 18160 comppfsc 23036 filunirn 23386 elflim 23475 flffval 23493 fclsval 23512 isfcls 23513 fcfval 23537 tsmsxplem1 23657 xmetunirn 23843 mopnval 23944 tmsval 23989 cfilfval 24781 caufval 24792 issgon 33121 elrnsiga 33124 volmeas 33229 omssubadd 33299 neibastop2lem 35245 ctbssinf 36287 ismtyval 36668 dicval 40047 prjcrv0 41375 ismrc 41439 nacsfix 41450 hbt 41872 |
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