Theorem ovprc 5923

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 5891	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		opprc 3811	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
3		0ex 4142	. . . 4 |- (/) e. _V
4	2, 3	eqeltrdi 2278	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		df-br 4016	. . . . 5 |- ( A dom F B <-> <. A , B >. e. dom F )
6		ovprc1.1	. . . . . 6 |- Rel dom F
7		brrelex12 4676	. . . . . 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 633	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> -. <. A , B >. e. dom F )
11		ndmfvg 5558	. . 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 2232	1 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   _Vcvv 2749   (/)c0 3434   <.cop 3607   class class class wbr 4015   dom cdm 4638   Rel wrel 4643   ` cfv 5228  (class class class)co 5888

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-dm 4648  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by:  ovprc1  5924  ovprc2  5925