Theorem predep 5706

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 5680	. 2 |- Pred ( _E , A , X ) = ( A i^i ( `' _E " { X } ) )
2		relcnv 5503	. . . . 5 |- Rel `' _E
3		relimasn 5488	. . . . 5 |- ( Rel `' _E -> ( `' _E " { X } ) = { y | X `' _E y } )
4	2, 3	ax-mp 5	. . . 4 |- ( `' _E " { X } ) = { y | X `' _E y }
5		vex 3203	. . . . . . 7 |- y e. _V
6		brcnvg 5303	. . . . . . 7 |- ( ( X e. B /\ y e. _V ) -> ( X `' _E y <-> y _E X ) )
7	5, 6	mpan2 707	. . . . . 6 |- ( X e. B -> ( X `' _E y <-> y _E X ) )
8		epelg 5030	. . . . . 6 |- ( X e. B -> ( y _E X <-> y e. X ) )
9	7, 8	bitrd 268	. . . . 5 |- ( X e. B -> ( X `' _E y <-> y e. X ) )
10	9	abbi1dv 2743	. . . 4 |- ( X e. B -> { y | X `' _E y } = X )
11	4, 10	syl5eq 2668	. . 3 |- ( X e. B -> ( `' _E " { X } ) = X )
12	11	ineq2d 3814	. 2 |- ( X e. B -> ( A i^i ( `' _E " { X } ) ) = ( A i^i X ) )
13	1, 12	syl5eq 2668	1 |- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   i^i cin 3573   {csn 4177   class class class wbr 4653   _E cep 5028   `'ccnv 5113   "cima 5117   Rel wrel 5119   Predcpred 5679

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-br 4654  df-opab 4713  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  predon  6991  omsinds  7084