Theorem brovmptimex 38325

Description: If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

Hypotheses

Ref	Expression
brovmptimex.mpt	|- F = ( x e. E , y e. G |-> H )
brovmptimex.br	|- ( ph -> A R B )
brovmptimex.ov	|- ( ph -> R = ( C F D ) )

Assertion

Ref	Expression
brovmptimex	|- ( ph -> ( C e. _V /\ D e. _V ) )

Distinct variable groups:   x, E, y   y, F

Allowed substitution hints:   ph( x, y)   A( x, y)   B( x, y)   C( x, y)   D( x, y)   R( x, y)   F( x)   G( x, y)   H( x, y)

Proof of Theorem brovmptimex

Step	Hyp	Ref	Expression
1		brovmptimex.ov	. . 3 |- ( ph -> R = ( C F D ) )
2		brovmptimex.br	. . 3 |- ( ph -> A R B )
3	1, 2	breqdi 4668	. 2 |- ( ph -> A ( C F D ) B )
4		brne0 4702	. 2 |- ( A ( C F D ) B -> ( C F D ) =/= (/) )
5		brovmptimex.mpt	. . . . 5 |- F = ( x e. E , y e. G |-> H )
6	5	reldmmpt2 6771	. . . 4 |- Rel dom F
7	6	ovprc 6683	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> ( C F D ) = (/) )
8	7	necon1ai 2821	. 2 |- ( ( C F D ) =/= (/) -> ( C e. _V /\ D e. _V ) )
9	3, 4, 8	3syl 18	1 |- ( ph -> ( C e. _V /\ D e. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  brovmptimex1  38326  brovmptimex2  38327