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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.) |
Ref | Expression |
---|---|
fvssunirn | ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvrn0 6691 | . . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | |
2 | elssuni 4859 | . . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}) |
4 | uniun 4849 | . . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅}) | |
5 | 0ex 5202 | . . . . 5 ⊢ ∅ ∈ V | |
6 | 5 | unisn 4846 | . . . 4 ⊢ ∪ {∅} = ∅ |
7 | 6 | uneq2i 4133 | . . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅) |
8 | un0 4341 | . . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹 | |
9 | 4, 7, 8 | 3eqtri 2845 | . 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹 |
10 | 3, 9 | sseqtri 4000 | 1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ cuni 4830 ran crn 5549 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 df-iota 6307 df-fv 6356 |
This theorem is referenced by: ovssunirn 7181 marypha2lem1 8887 acnlem 9462 fin23lem29 9751 itunitc 9831 hsmexlem5 9840 wunfv 10142 wunex2 10148 strfvss 16494 prdsval 16716 prdsbas 16718 prdsplusg 16719 prdsmulr 16720 prdsvsca 16721 prdshom 16728 mreunirn 16860 mrcfval 16867 mrcssv 16873 mrisval 16889 sscpwex 17073 wunfunc 17157 catcxpccl 17445 comppfsc 22068 filunirn 22418 elflim 22507 flffval 22525 fclsval 22544 isfcls 22545 fcfval 22569 tsmsxplem1 22688 xmetunirn 22874 mopnval 22975 tmsval 23018 cfilfval 23794 caufval 23805 issgon 31281 elrnsiga 31284 volmeas 31389 omssubadd 31457 neibastop2lem 33605 ctbssinf 34569 ismtyval 34959 dicval 38192 ismrc 39176 nacsfix 39187 hbt 39608 |
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