Theorem oprprc1 3019

Description: The value of an operation when the first argument is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair.

Hypothesis

Ref	Expression
oprprc1.1	⊢ Rel dom F

Assertion

Ref	Expression
oprprc1	⊢ (¬ A ∈ V → (AFB) = ∅)

Proof of Theorem oprprc1

Step	Hyp	Ref	Expression
1		df-br 2063	. . . . 5 ⊢ (Adom FB ↔ ⟨A, B⟩ ∈ dom F)
2		oprprc1.1	. . . . . 6 ⊢ Rel dom F
3	2	brrelexi 2447	. . . . 5 ⊢ (Adom FB → A ∈ V)
4	1, 3	sylbir 176	. . . 4 ⊢ (⟨A, B⟩ ∈ dom F → A ∈ V)
5	4	con3i 90	. . 3 ⊢ (¬ A ∈ V → ¬ ⟨A, B⟩ ∈ dom F)
6		ndmfv 2848	. . 3 ⊢ (¬ ⟨A, B⟩ ∈ dom F → (F ‘⟨A, B⟩) = ∅)
7	5, 6	syl 12	. 2 ⊢ (¬ A ∈ V → (F ‘⟨A, B⟩) = ∅)
8		df-opr 3003	. 2 ⊢ (AFB) = (F ‘⟨A, B⟩)
9	7, 8	syl5eq 1136	1 ⊢ (¬ A ∈ V → (AFB) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  ⟨cop 1810   class class class wbr 2054  dom cdm 2410  Rel wrel 2415   ‘cfv 2422  (class class class)co 3001

This theorem is referenced by:  ndmoprcl 3058

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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