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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 6870 | . 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹) | |
2 | 1 | ssriv 3947 | 1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3909 ∪ cuni 4864 ran crn 5632 ‘cfv 6492 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-cnv 5639 df-dm 5641 df-rn 5642 df-iota 6444 df-fv 6500 |
This theorem is referenced by: ovssunirn 7386 marypha2lem1 9305 acnlem 9918 fin23lem29 10211 itunitc 10291 hsmexlem5 10300 wunfv 10602 wunex2 10608 strfvss 16995 prdsvallem 17272 prdsval 17273 prdsbas 17275 prdsplusg 17276 prdsmulr 17277 prdsvsca 17278 prdshom 17285 mreunirn 17417 mrcfval 17424 mrcssv 17430 mrisval 17446 sscpwex 17634 wunfunc 17721 wunfuncOLD 17722 catcxpccl 18031 catcxpcclOLD 18032 comppfsc 22811 filunirn 23161 elflim 23250 flffval 23268 fclsval 23287 isfcls 23288 fcfval 23312 tsmsxplem1 23432 xmetunirn 23618 mopnval 23719 tmsval 23764 cfilfval 24556 caufval 24567 issgon 32502 elrnsiga 32505 volmeas 32610 omssubadd 32680 neibastop2lem 34763 ctbssinf 35808 ismtyval 36190 dicval 39570 prjcrv0 40873 ismrc 40926 nacsfix 40937 hbt 41359 |
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