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Theorem ovprc 5459
 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 5435 . 2 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 opprc 3540 . . . 4 (¬ (A V B V) → ⟨A, B⟩ = ∅)
3 0ex 3854 . . . 4 V
42, 3syl6eqel 2106 . . 3 (¬ (A V B V) → ⟨A, B V)
5 df-br 3735 . . . . 5 (Adom 𝐹 B ↔ ⟨A, B dom 𝐹)
6 ovprc1.1 . . . . . 6 Rel dom 𝐹
7 brrelex12 4304 . . . . . 6 ((Rel dom 𝐹 Adom 𝐹 B) → (A V B V))
86, 7mpan 402 . . . . 5 (Adom 𝐹 B → (A V B V))
95, 8sylbir 125 . . . 4 (⟨A, B dom 𝐹 → (A V B V))
109con3i 549 . . 3 (¬ (A V B V) → ¬ ⟨A, B dom 𝐹)
11 ndmfvg 5125 . . 3 ((⟨A, B V ¬ ⟨A, B dom 𝐹) → (𝐹‘⟨A, B⟩) = ∅)
124, 10, 11syl2anc 393 . 2 (¬ (A V B V) → (𝐹‘⟨A, B⟩) = ∅)
131, 12syl5eq 2062 1 (¬ (A V B V) → (A𝐹B) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  Vcvv 2531  ∅c0 3197  ⟨cop 3349   class class class wbr 3734  dom cdm 4268  Rel wrel 4273  ‘cfv 4825  (class class class)co 5432 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-dm 4278  df-iota 4790  df-fv 4833  df-ov 5435 This theorem is referenced by:  ovprc1  5460  ovprc2  5461
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