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Theorem ovprc 5856
Description: The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
ovprc1.1 Rel dom 𝐹
Assertion
Ref Expression
ovprc (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem ovprc
StepHypRef Expression
1 df-ov 5827 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 opprc 3762 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3 0ex 4091 . . . 4 ∅ ∈ V
42, 3eqeltrdi 2248 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
5 df-br 3966 . . . . 5 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ovprc1.1 . . . . . 6 Rel dom 𝐹
7 brrelex12 4624 . . . . . 6 ((Rel dom 𝐹𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
86, 7mpan 421 . . . . 5 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
95, 8sylbir 134 . . . 4 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
109con3i 622 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
11 ndmfvg 5499 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
124, 10, 11syl2anc 409 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
131, 12syl5eq 2202 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wcel 2128  Vcvv 2712  c0 3394  cop 3563   class class class wbr 3965  dom cdm 4586  Rel wrel 4591  cfv 5170  (class class class)co 5824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-dm 4596  df-iota 5135  df-fv 5178  df-ov 5827
This theorem is referenced by:  ovprc1  5857  ovprc2  5858
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