Theorem ovssunirn 7452

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 7419	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		fvssunirn 6925	. 2 ⊢ (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ∪ ran 𝐹
3	1, 2	eqsstri 4007	1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3939  ⟨cop 4630  ∪ cuni 4903  ran crn 5673  ‘cfv 6543  (class class class)co 7416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-iota 6495  df-fv 6551  df-ov 7419

This theorem is referenced by:  prdsvallem  17435  prdsplusg  17439  prdsmulr  17440  prdsvsca  17441  prdshom  17448  wunfunc  17886  wunfuncOLD  17887  wunnat  17945  wunnatOLD  17946  homarw  18034  catcoppccl  18105  catcoppcclOLD  18106  catcfuccl  18107  catcfucclOLD  18108  catcxpccl  18197  catcxpcclOLD  18198  isanmbfmOLD  33931  isanmbfm  33933