Theorem oprprc2 3020

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

Assertion

Ref	Expression
oprprc2	⊢ (¬ B ∈ V → (AFB) = (AFA))

Proof of Theorem oprprc2

Step	Hyp	Ref	Expression
1		opprc2 1907	. . 3 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
2	1	fveq2d 2836	. 2 ⊢ (¬ B ∈ V → (F ‘⟨A, B⟩) = (F ‘⟨A, A⟩))
3		df-opr 3003	. 2 ⊢ (AFB) = (F ‘⟨A, B⟩)
4		df-opr 3003	. 2 ⊢ (AFA) = (F ‘⟨A, A⟩)
5	2, 3, 4	3eqtr4g 1147	1 ⊢ (¬ B ∈ V → (AFB) = (AFA))

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   ‘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-clab 1093  df-cleq 1097  df-clel 1099  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-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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