Theorem ovprc 6683

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 e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Proof of Theorem ovprc

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		df-br 4654	. . . . 5 |- ( A dom F B <-> <. A , B >. e. dom F )
3		ovprc1.1	. . . . . 6 |- Rel dom F
4		brrelex12 5155	. . . . . 6 |- ( ( Rel dom F /\ A dom F B ) -> ( A e. _V /\ B e. _V ) )
5	3, 4	mpan 706	. . . . 5 |- ( A dom F B -> ( A e. _V /\ B e. _V ) )
6	2, 5	sylbir 225	. . . 4 |- ( <. A , B >. e. dom F -> ( A e. _V /\ B e. _V ) )
7	6	con3i 150	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> -. <. A , B >. e. dom F )
8		ndmfv 6218	. . 3 |- ( -. <. A , B >. e. dom F -> ( F ` <. A , B >. ) = (/) )
9	7, 8	syl 17	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> ( F ` <. A , B >. ) = (/) )
10	1, 9	syl5eq 2668	1 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   dom cdm 5114   Rel wrel 5119   ` cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ovprc1  6684  ovprc2  6685  ovrcl  6686  elbasov  15921  firest  16093  psrplusg  19381  psrmulr  19384  psrvscafval  19390  mplval  19428  opsrle  19475  opsrbaslem  19477  opsrbaslemOLD  19478  evlval  19524  matbas0pc  20215  mdetfval  20392  madufval  20443  mdegfval  23822  nbgrprc0  26229  brovmptimex  38325