Theorem brovex 7348

Description: A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)

Hypotheses

Ref	Expression
brovex.1	|- O = ( x e. _V , y e. _V |-> C )
brovex.2	|- ( ( V e. _V /\ E e. _V ) -> Rel ( V O E ) )

Assertion

Ref	Expression
brovex	|- ( F ( V O E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   C( x, y)   P( x, y)   E( x, y)   F( x, y)   O( x, y)   V( x, y)

Proof of Theorem brovex

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 |- ( F ( V O E ) P <-> <. F , P >. e. ( V O E ) )
2		ne0i 3921	. . . 4 |- ( <. F , P >. e. ( V O E ) -> ( V O E ) =/= (/) )
3		brovex.1	. . . . . 6 |- O = ( x e. _V , y e. _V |-> C )
4	3	mpt2ndm0 6875	. . . . 5 |- ( -. ( V e. _V /\ E e. _V ) -> ( V O E ) = (/) )
5	4	necon1ai 2821	. . . 4 |- ( ( V O E ) =/= (/) -> ( V e. _V /\ E e. _V ) )
6		brovex.2	. . . . . . 7 |- ( ( V e. _V /\ E e. _V ) -> Rel ( V O E ) )
7		brrelex12 5155	. . . . . . 7 |- ( ( Rel ( V O E ) /\ F ( V O E ) P ) -> ( F e. _V /\ P e. _V ) )
8	6, 7	sylan 488	. . . . . 6 |- ( ( ( V e. _V /\ E e. _V ) /\ F ( V O E ) P ) -> ( F e. _V /\ P e. _V ) )
9		id 22	. . . . . 6 |- ( ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
10	8, 9	syldan 487	. . . . 5 |- ( ( ( V e. _V /\ E e. _V ) /\ F ( V O E ) P ) -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
11	10	ex 450	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( F ( V O E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) ) )
12	2, 5, 11	3syl 18	. . 3 |- ( <. F , P >. e. ( V O E ) -> ( F ( V O E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) ) )
13	1, 12	sylbi 207	. 2 |- ( F ( V O E ) P -> ( F ( V O E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) ) )
14	13	pm2.43i 52	1 |- ( F ( V O E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   Rel wrel 5119  (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:  brovmpt2ex  7349