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Theorem ovprc 5482
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 5458 . 2 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 opprc 3561 . . . 4 (¬ (A V B V) → ⟨A, B⟩ = ∅)
3 0ex 3875 . . . 4 V
42, 3syl6eqel 2125 . . 3 (¬ (A V B V) → ⟨A, B V)
5 df-br 3756 . . . . 5 (Adom 𝐹 B ↔ ⟨A, B dom 𝐹)
6 ovprc1.1 . . . . . 6 Rel dom 𝐹
7 brrelex12 4324 . . . . . 6 ((Rel dom 𝐹 Adom 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 561 . . 3 (¬ (A V B V) → ¬ ⟨A, B dom 𝐹)
11 ndmfvg 5147 . . 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 2081 1 (¬ (A V B V) → (A𝐹B) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  c0 3218  cop 3370   class class class wbr 3755  dom cdm 4288  Rel wrel 4293  cfv 4845  (class class class)co 5455
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  ovprc1  5483  ovprc2  5484
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