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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 7138 | . 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
2 | fvssunirn 6674 | . 2 ⊢ (𝐹‘〈𝑋, 𝑌〉) ⊆ ∪ ran 𝐹 | |
3 | 1, 2 | eqsstri 3949 | 1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3881 〈cop 4531 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 (class class class)co 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: prdsval 16720 prdsplusg 16723 prdsmulr 16724 prdsvsca 16725 prdshom 16732 wunfunc 17161 wunnat 17218 homarw 17298 catcoppccl 17360 catcfuccl 17361 catcxpccl 17449 |
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