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Mirrors > Home > MPE Home > Th. List > ovssunirn | Structured version Visualization version GIF version |
Description: The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
ovssunirn | ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7148 | . 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
2 | fvssunirn 6693 | . 2 ⊢ (𝐹‘〈𝑋, 𝑌〉) ⊆ ∪ ran 𝐹 | |
3 | 1, 2 | eqsstri 4000 | 1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3935 〈cop 4565 ∪ cuni 4832 ran crn 5550 ‘cfv 6349 (class class class)co 7145 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 df-iota 6308 df-fv 6357 df-ov 7148 |
This theorem is referenced by: prdsval 16718 prdsplusg 16721 prdsmulr 16722 prdsvsca 16723 prdshom 16730 wunfunc 17159 wunnat 17216 homarw 17296 catcoppccl 17358 catcfuccl 17359 catcxpccl 17447 |
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