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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 6216 | . . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | |
2 | elssuni 4467 | . . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}) |
4 | uniun 4456 | . . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅}) | |
5 | 0ex 4790 | . . . . 5 ⊢ ∅ ∈ V | |
6 | 5 | unisn 4451 | . . . 4 ⊢ ∪ {∅} = ∅ |
7 | 6 | uneq2i 3764 | . . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅) |
8 | un0 3967 | . . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹 | |
9 | 4, 7, 8 | 3eqtri 2648 | . 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹 |
10 | 3, 9 | sseqtri 3637 | 1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 ∪ cuni 4436 ran crn 5115 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 |
This theorem is referenced by: ovssunirn 6681 marypha2lem1 8341 acnlem 8871 fin23lem29 9163 itunitc 9243 hsmexlem5 9252 wunfv 9554 wunex2 9560 strfvss 15880 prdsval 16115 prdsbas 16117 prdsplusg 16118 prdsmulr 16119 prdsvsca 16120 prdshom 16127 mreunirn 16261 mrcfval 16268 mrcssv 16274 mrisval 16290 sscpwex 16475 wunfunc 16559 catcxpccl 16847 comppfsc 21335 filunirn 21686 elflim 21775 flffval 21793 fclsval 21812 isfcls 21813 fcfval 21837 tsmsxplem1 21956 xmetunirn 22142 mopnval 22243 tmsval 22286 cfilfval 23062 caufval 23073 issgon 30186 elrnsiga 30189 volmeas 30294 omssubadd 30362 neibastop2lem 32355 ismtyval 33599 dicval 36465 ismrc 37264 nacsfix 37275 hbt 37700 |
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