Theorem predep 13903

Description: The predecessor under the epsilon relationship is equivalent to an intersection. (The proof was 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

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . 7 |- y e. _V
2		brcnvg 4142	. . . . . . 7 |- ((X e. B /\ y e. _V) -> (X`' _E y <-> y _E X))
3	1, 2	mpan2 760	. . . . . 6 |- (X e. B -> (X`' _E y <-> y _E X))
4		epelg 13827	. . . . . 6 |- (X e. B -> (y _E X <-> y e. X))
5	3, 4	bitrd 587	. . . . 5 |- (X e. B -> (X`' _E y <-> y e. X))
6	5	abbi1dv 2010	. . . 4 |- (X e. B -> {y | X`' _E y} = X)
7		relcnv 4301	. . . . 5 |- Rel `' _E
8		relimasn 4288	. . . . 5 |- (Rel `' _E -> (`' _E "{X}) = {y | X`' _E y})
9	7, 8	ax-mp 7	. . . 4 |- (`' _E "{X}) = {y | X`' _E y}
10	6, 9	syl5eq 1940	. . 3 |- (X e. B -> (`' _E "{X}) = X)
11	10	ineq2d 2796	. 2 |- (X e. B -> (A i^i (`' _E "{X})) = (A i^i X))
12		df-pred 13880	. 2 |- Pred( _E , A, X) = (A i^i (`' _E "{X}))
13	11, 12	syl5eq 1940	1 |- (X e. B -> Pred( _E , A, X) = (A i^i X))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   i^i cin 2592  {csn 3044   class class class wbr 3338   _E cep 3581  `'ccnv 3985  "cima 3989  Rel wrel 3991  Predcpred 13879

This theorem is referenced by:  predon 13904  epsetlike 13905

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-pred 13880

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