Theorem ovssunirn 7394

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 7361	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		fvssunirn 6864	. 2 ⊢ (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ∪ ran 𝐹
3	1, 2	eqsstri 3979	1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3900  ⟨cop 4585  ∪ cuni 4862  ran crn 5624  ‘cfv 6491  (class class class)co 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-cnv 5631  df-dm 5633  df-rn 5634  df-iota 6447  df-fv 6499  df-ov 7361

This theorem is referenced by:  prdsvallem  17376  prdsplusg  17380  prdsmulr  17381  prdsvsca  17382  prdshom  17389  wunfunc  17827  wunnat  17885  homarw  17972  catcoppccl  18043  catcfuccl  18044  catcxpccl  18132  isanmbfm  34392