Theorem ovprc 6054

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 6021	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		opprc 3883	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
3		0ex 4216	. . . 4 |- (/) e. _V
4	2, 3	eqeltrdi 2322	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		df-br 4089	. . . . 5 |- ( A dom F B <-> <. A , B >. e. dom F )
6		ovprc1.1	. . . . . 6 |- Rel dom F
7		brrelex12 4764	. . . . . 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 637	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> -. <. A , B >. e. dom F )
11		ndmfvg 5670	. . 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 2276	1 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   (/)c0 3494   <.cop 3672   class class class wbr 4088   dom cdm 4725   Rel wrel 4730   ` cfv 5326  (class class class)co 6018

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-dm 4735  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by:  ovprc1  6055  ovprc2  6056