Theorem ovssunirng 6087

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
ovssunirng	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹)

Proof of Theorem ovssunirng

Step	Hyp	Ref	Expression
1		df-ov 6055	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		opexg 4346	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ V)
3		fvssunirng 5687	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ V → (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ∪ ran 𝐹)
4	2, 3	syl 14	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ∪ ran 𝐹)
5	1, 4	eqsstrid 3286	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3213  ⟨cop 3694  ∪ cuni 3916  ran crn 4752  ‘cfv 5354  (class class class)co 6052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-cnv 4759  df-dm 4761  df-rn 4762  df-iota 5314  df-fv 5362  df-ov 6055

This theorem is referenced by:  prdsvallem  13506