Theorem ovssunirng 6058

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 6026	. 2 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2		opexg 4322	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ V)
3		fvssunirng 5657	. . 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 3272	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2201  Vcvv 2801   ⊆ wss 3199  ⟨cop 3673  ∪ cuni 3894  ran crn 4728  ‘cfv 5328  (class class class)co 6023

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-cnv 4735  df-dm 4737  df-rn 4738  df-iota 5288  df-fv 5336  df-ov 6026

This theorem is referenced by:  prdsvallem  13378