Theorem predep 25414

Description: The predecessor under the epsilon relationship is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
predep	|- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) )

Proof of Theorem predep

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pred 25390	. 2 |- Pred ( _E , A , X ) = ( A i^i ( `' _E " { X } ) )
2		relcnv 5209	. . . . 5 |- Rel `' _E
3		relimasn 5194	. . . . 5 |- ( Rel `' _E -> ( `' _E " { X } ) = { y | X `' _E y } )
4	2, 3	ax-mp 8	. . . 4 |- ( `' _E " { X } ) = { y | X `' _E y }
5		vex 2927	. . . . . . 7 |- y e. _V
6		brcnvg 5020	. . . . . . 7 |- ( ( X e. B /\ y e. _V ) -> ( X `' _E y <-> y _E X ) )
7	5, 6	mpan2 653	. . . . . 6 |- ( X e. B -> ( X `' _E y <-> y _E X ) )
8		epelg 4463	. . . . . 6 |- ( X e. B -> ( y _E X <-> y e. X ) )
9	7, 8	bitrd 245	. . . . 5 |- ( X e. B -> ( X `' _E y <-> y e. X ) )
10	9	abbi1dv 2528	. . . 4 |- ( X e. B -> { y | X `' _E y } = X )
11	4, 10	syl5eq 2456	. . 3 |- ( X e. B -> ( `' _E " { X } ) = X )
12	11	ineq2d 3510	. 2 |- ( X e. B -> ( A i^i ( `' _E " { X } ) ) = ( A i^i X ) )
13	1, 12	syl5eq 2456	1 |- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) )

Syntax hints:   -> wi 4   <-> wb 177   = wceq 1649   e. wcel 1721   {cab 2398   _Vcvv 2924   i^i cin 3287   {csn 3782   class class class wbr 4180   _E cep 4460   `'ccnv 4844   "cima 4848   Rel wrel 4850   Predcpred 25389

This theorem is referenced by:  predon  25415  epsetlike  25416  omsinds  25441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-eprel 4462  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 25390