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Theorem ovprc 5540
 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 5515 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 opprc 3570 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3 0ex 3884 . . . 4 ∅ ∈ V
42, 3syl6eqel 2128 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
5 df-br 3765 . . . . 5 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ovprc1.1 . . . . . 6 Rel dom 𝐹
7 brrelex12 4381 . . . . . 6 ((Rel dom 𝐹𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
86, 7mpan 400 . . . . 5 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
95, 8sylbir 125 . . . 4 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
109con3i 562 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
11 ndmfvg 5204 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
124, 10, 11syl2anc 391 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
131, 12syl5eq 2084 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  ⟨cop 3378   class class class wbr 3764  dom cdm 4345  Rel wrel 4350  ‘cfv 4902  (class class class)co 5512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  ovprc1  5541  ovprc2  5542
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