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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 16994 prdsvallem 17271 prdsval 17272 prdsbas 17274 prdsplusg 17275 prdsmulr 17276 prdsvsca 17277 prdshom 17284 mreunirn 17416 mrcfval 17423 mrcssv 17429 mrisval 17445 sscpwex 17633 wunfunc 17720 wunfuncOLD 17721 catcxpccl 18030 catcxpcclOLD 18031 comppfsc 22806 filunirn 23156 elflim 23245 flffval 23263 fclsval 23282 isfcls 23283 fcfval 23307 tsmsxplem1 23427 xmetunirn 23613 mopnval 23714 tmsval 23759 cfilfval 24551 caufval 24562 issgon 32496 elrnsiga 32499 volmeas 32604 omssubadd 32674 neibastop2lem 34728 ctbssinf 35773 ismtyval 36155 dicval 39535 prjcrv0 40837 ismrc 40890 nacsfix 40901 hbt 41323 |
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