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 6913	. 2 ⊢ (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ∪ ran 𝐹
3	1, 2	eqsstri 3980	1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹

This proof depends on syntax axioms:   ⊆ wss 3902  ⟨cop 4593  ∪ cuni 4870  ran crn 5660  ‘cfv 6537  (class class class)co 7416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  prdsvallem  17543  prdsplusg  17547  prdsmulr  17548  prdsvsca  17549  prdshom  17556  wunfunc  17994  wunnat  18052  homarw  18139  catcoppccl  18210  catcfuccl  18211  catcxpccl  18299  isanmbfm  34754