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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 7415 | . 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
2 | fvssunirn 6924 | . 2 ⊢ (𝐹‘〈𝑋, 𝑌〉) ⊆ ∪ ran 𝐹 | |
3 | 1, 2 | eqsstri 4016 | 1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3948 〈cop 4634 ∪ cuni 4908 ran crn 5677 ‘cfv 6543 (class class class)co 7412 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-iota 6495 df-fv 6551 df-ov 7415 |
This theorem is referenced by: prdsvallem 17407 prdsplusg 17411 prdsmulr 17412 prdsvsca 17413 prdshom 17420 wunfunc 17858 wunfuncOLD 17859 wunnat 17917 wunnatOLD 17918 homarw 18006 catcoppccl 18077 catcoppcclOLD 18078 catcfuccl 18079 catcfucclOLD 18080 catcxpccl 18169 catcxpcclOLD 18170 isanmbfmOLD 33716 isanmbfm 33718 |
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