Theorem ovprc 6037

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 6004	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		opprc 3878	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
3		0ex 4211	. . . 4 |- (/) e. _V
4	2, 3	eqeltrdi 2320	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		df-br 4084	. . . . 5 |- ( A dom F B <-> <. A , B >. e. dom F )
6		ovprc1.1	. . . . . 6 |- Rel dom F
7		brrelex12 4757	. . . . . 6 |- ( ( Rel dom F /\ A dom F B ) -> ( A e. _V /\ B e. _V ) )
8	6, 7	mpan 424	. . . . 5 |- ( A dom F B -> ( A e. _V /\ B e. _V ) )
9	5, 8	sylbir 135	. . . 4 |- ( <. A , B >. e. dom F -> ( A e. _V /\ B e. _V ) )
10	9	con3i 635	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> -. <. A , B >. e. dom F )
11		ndmfvg 5658	. . 3 |- ( ( <. A , B >. e. _V /\ -. <. A , B >. e. dom F ) -> ( F ` <. A , B >. ) = (/) )
12	4, 10, 11	syl2anc 411	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> ( F ` <. A , B >. ) = (/) )
13	1, 12	eqtrid 2274	1 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   (/)c0 3491   <.cop 3669   class class class wbr 4083   dom cdm 4719   Rel wrel 4724   ` cfv 5318  (class class class)co 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-dm 4729  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  ovprc1  6038  ovprc2  6039