Theorem ovssunirng 5960

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 5928	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		opexg 4262	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ V)
3		fvssunirng 5576	. . 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 3230	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  ⟨cop 3626  ∪ cuni 3840  ran crn 4665  ‘cfv 5259  (class class class)co 5925

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-cnv 4672  df-dm 4674  df-rn 4675  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  prdsvallem  12974