Theorem brovmpt2ex 7349

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.)

Hypothesis

Ref	Expression
brovmpt2ex.1	|- O = ( x e. _V , y e. _V |-> { <. z , w >. | ph } )

Assertion

Ref	Expression
brovmpt2ex	|- ( F ( V O E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )

Distinct variable group:   x, w, y, z

Allowed substitution hints:   ph( x, y, z, w)   P( x, y, z, w)   E( x, y, z, w)   F( x, y, z, w)   O( x, y, z, w)   V( x, y, z, w)

Proof of Theorem brovmpt2ex

Step	Hyp	Ref	Expression
1		brovmpt2ex.1	. 2 |- O = ( x e. _V , y e. _V |-> { <. z , w >. | ph } )
2	1	relmpt2opab 7259	. . 3 |- Rel ( V O E )
3	2	a1i 11	. 2 |- ( ( V e. _V /\ E e. _V ) -> Rel ( V O E ) )
4	1, 3	brovex 7348	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   _Vcvv 3200   class class class wbr 4653   {copab 4712   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  ax-un 6949

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-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)