Theorem ovssunirng 5986

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 5954	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		opexg 4276	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ V)
3		fvssunirng 5598	. . 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 3240	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2177  Vcvv 2773   ⊆ wss 3167  ⟨cop 3637  ∪ cuni 3852  ran crn 4680  ‘cfv 5276  (class class class)co 5951

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-cnv 4687  df-dm 4689  df-rn 4690  df-iota 5237  df-fv 5284  df-ov 5954

This theorem is referenced by:  prdsvallem  13148