Theorem predep 25472

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 25444	. 2 |- Pred ( _E , A , X ) = ( A i^i ( `' _E " { X } ) )
2		relcnv 5245	. . . . 5 |- Rel `' _E
3		relimasn 5230	. . . . 5 |- ( Rel `' _E -> ( `' _E " { X } ) = { y | X `' _E y } )
4	2, 3	ax-mp 5	. . . 4 |- ( `' _E " { X } ) = { y | X `' _E y }
5		vex 2961	. . . . . . 7 |- y e. _V
6		brcnvg 5056	. . . . . . 7 |- ( ( X e. B /\ y e. _V ) -> ( X `' _E y <-> y _E X ) )
7	5, 6	mpan2 654	. . . . . 6 |- ( X e. B -> ( X `' _E y <-> y _E X ) )
8		epelg 4498	. . . . . 6 |- ( X e. B -> ( y _E X <-> y e. X ) )
9	7, 8	bitrd 246	. . . . 5 |- ( X e. B -> ( X `' _E y <-> y e. X ) )
10	9	abbi1dv 2554	. . . 4 |- ( X e. B -> { y | X `' _E y } = X )
11	4, 10	syl5eq 2482	. . 3 |- ( X e. B -> ( `' _E " { X } ) = X )
12	11	ineq2d 3544	. 2 |- ( X e. B -> ( A i^i ( `' _E " { X } ) ) = ( A i^i X ) )
13	1, 12	syl5eq 2482	1 |- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) )

Syntax hints:   -> wi 4   <-> wb 178   = wceq 1653   e. wcel 1726   {cab 2424   _Vcvv 2958   i^i cin 3321   {csn 3816   class class class wbr 4215   _E cep 4495   `'ccnv 4880   "cima 4884   Rel wrel 4886   Predcpred 25443

This theorem is referenced by:  predon  25473  epsetlike  25474  omsinds  25499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-pred 25444