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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 7161 | . 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
2 | fvssunirn 6701 | . 2 ⊢ (𝐹‘〈𝑋, 𝑌〉) ⊆ ∪ ran 𝐹 | |
3 | 1, 2 | eqsstri 4003 | 1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3938 〈cop 4575 ∪ cuni 4840 ran crn 5558 ‘cfv 6357 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-cnv 5565 df-dm 5567 df-rn 5568 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: prdsval 16730 prdsplusg 16733 prdsmulr 16734 prdsvsca 16735 prdshom 16742 wunfunc 17171 wunnat 17228 homarw 17308 catcoppccl 17370 catcfuccl 17371 catcxpccl 17459 |
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