Theorem predep 24192

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 24168	. 2 |- Pred ( _E , A , X ) = ( A i^i ( `' _E " { X } ) )
2		relcnv 5051	. . . . 5 |- Rel `' _E
3		relimasn 5036	. . . . 5 |- ( Rel `' _E -> ( `' _E " { X } ) = { y | X `' _E y } )
4	2, 3	ax-mp 8	. . . 4 |- ( `' _E " { X } ) = { y | X `' _E y }
5		vex 2791	. . . . . . 7 |- y e. _V
6		brcnvg 4862	. . . . . . 7 |- ( ( X e. B /\ y e. _V ) -> ( X `' _E y <-> y _E X ) )
7	5, 6	mpan2 652	. . . . . 6 |- ( X e. B -> ( X `' _E y <-> y _E X ) )
8		epelg 4306	. . . . . 6 |- ( X e. B -> ( y _E X <-> y e. X ) )
9	7, 8	bitrd 244	. . . . 5 |- ( X e. B -> ( X `' _E y <-> y e. X ) )
10	9	abbi1dv 2399	. . . 4 |- ( X e. B -> { y | X `' _E y } = X )
11	4, 10	syl5eq 2327	. . 3 |- ( X e. B -> ( `' _E " { X } ) = X )
12	11	ineq2d 3370	. 2 |- ( X e. B -> ( A i^i ( `' _E " { X } ) ) = ( A i^i X ) )
13	1, 12	syl5eq 2327	1 |- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) )

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1623   e. wcel 1684   {cab 2269   _Vcvv 2788   i^i cin 3151   {csn 3640   class class class wbr 4023   _E cep 4303   `'ccnv 4688   "cima 4692   Rel wrel 4694   Predcpred 24167

This theorem is referenced by:  predon  24193  epsetlike  24194  omsinds  24219

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168