Theorem ovprc 6003

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 5970	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		opprc 3854	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
3		0ex 4187	. . . 4 |- (/) e. _V
4	2, 3	eqeltrdi 2298	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		df-br 4060	. . . . 5 |- ( A dom F B <-> <. A , B >. e. dom F )
6		ovprc1.1	. . . . . 6 |- Rel dom F
7		brrelex12 4731	. . . . . 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 5630	. . 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 2252	1 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   (/)c0 3468   <.cop 3646   class class class wbr 4059   dom cdm 4693   Rel wrel 4698   ` cfv 5290  (class class class)co 5967

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-dm 4703  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  ovprc1  6004  ovprc2  6005