Theorem relfae 30310

Description: The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Assertion

Ref	Expression
relfae	|- ( ( R e. _V /\ M e. U. ran measures ) -> Rel ( R~ a.e. M ) )

Proof of Theorem relfae

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 5247	. 2 |- Rel { <. f , g >. | ( ( f e. ( dom R ^m U. dom M ) /\ g e. ( dom R ^m U. dom M ) ) /\ { x e. U. dom M | ( f ` x ) R ( g ` x ) }a.e. M ) }
2		faeval 30309	. . 3 |- ( ( R e. _V /\ M e. U. ran measures ) -> ( R~ a.e. M ) = { <. f , g >. | ( ( f e. ( dom R ^m U. dom M ) /\ g e. ( dom R ^m U. dom M ) ) /\ { x e. U. dom M | ( f ` x ) R ( g ` x ) }a.e. M ) } )
3	2	releqd 5203	. 2 |- ( ( R e. _V /\ M e. U. ran measures ) -> ( Rel ( R~ a.e. M ) <-> Rel { <. f , g >. | ( ( f e. ( dom R ^m U. dom M ) /\ g e. ( dom R ^m U. dom M ) ) /\ { x e. U. dom M | ( f ` x ) R ( g ` x ) }a.e. M ) } ) )
4	1, 3	mpbiri 248	1 |- ( ( R e. _V /\ M e. U. ran measures ) -> Rel ( R~ a.e. M ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {crab 2916   _Vcvv 3200   U.cuni 4436   class class class wbr 4653   {copab 4712   dom cdm 5114   ran crn 5115   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   ^m cmap 7857  measurescmeas 30258  a.e.cae 30300  ~ a.e.cfae 30301

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fae 30308

This theorem is referenced by: (None)