![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7434 | . 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
2 | fvssunirn 6940 | . 2 ⊢ (𝐹‘〈𝑋, 𝑌〉) ⊆ ∪ ran 𝐹 | |
3 | 1, 2 | eqsstri 4030 | 1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3963 〈cop 4637 ∪ cuni 4912 ran crn 5690 ‘cfv 6563 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: prdsvallem 17501 prdsplusg 17505 prdsmulr 17506 prdsvsca 17507 prdshom 17514 wunfunc 17952 wunfuncOLD 17953 wunnat 18011 wunnatOLD 18012 homarw 18100 catcoppccl 18171 catcoppcclOLD 18172 catcfuccl 18173 catcfucclOLD 18174 catcxpccl 18263 catcxpcclOLD 18264 isanmbfmOLD 34236 isanmbfm 34238 |
Copyright terms: Public domain | W3C validator |