Theorem ovssunirn 7448

Description: The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)

Assertion

Ref	Expression
ovssunirn	⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹

Proof of Theorem ovssunirn

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

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