Theorem ovprc 5980

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 5947	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		opprc 3840	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
3		0ex 4171	. . . 4 |- (/) e. _V
4	2, 3	eqeltrdi 2296	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		df-br 4045	. . . . 5 |- ( A dom F B <-> <. A , B >. e. dom F )
6		ovprc1.1	. . . . . 6 |- Rel dom F
7		brrelex12 4713	. . . . . 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 5607	. . 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 2250	1 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   (/)c0 3460   <.cop 3636   class class class wbr 4044   dom cdm 4675   Rel wrel 4680   ` cfv 5271  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-dm 4685  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  ovprc1  5981  ovprc2  5982