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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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