Theorem ovssunirn 7434

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 7401	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		fvssunirn 6900	. 2 ⊢ (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ∪ ran 𝐹
3	1, 2	eqsstri 3984	1 ⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3906  ⟨cop 4590  ∪ cuni 4867  ran crn 5650  ‘cfv 6523  (class class class)co 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660  df-iota 6479  df-fv 6531  df-ov 7401

This theorem is referenced by:  prdsvallem  17485  prdsplusg  17489  prdsmulr  17490  prdsvsca  17491  prdshom  17498  wunfunc  17936  wunnat  17994  homarw  18081  catcoppccl  18152  catcfuccl  18153  catcxpccl  18241  isanmbfm  34555