Theorem ovssunirng 6114

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 6082	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		opexg 4366	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ V)
3		fvssunirng 5708	. . 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 3294	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  ⟨cop 3711  ∪ cuni 3933  ran crn 4773  ‘cfv 5375  (class class class)co 6079

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-cnv 4780  df-dm 4782  df-rn 4783  df-iota 5335  df-fv 5383  df-ov 6082

This theorem is referenced by:  prdsvallem  13604