Theorem dfpred3 5690

Description: An alternate definition of predecessor class when X is a set. (Contributed by Scott Fenton, 13-Jun-2018.)

Hypothesis

Ref	Expression
dfpred2.1	|- X e. _V

Assertion

Ref	Expression
dfpred3	|- Pred ( R , A , X ) = { y e. A | y R X }

Distinct variable groups:   y, R   y, X   y, A

Proof of Theorem dfpred3

Step	Hyp	Ref	Expression
1		incom 3805	. 2 |- ( A i^i { y | y R X } ) = ( { y | y R X } i^i A )
2		dfpred2.1	. . 3 |- X e. _V
3	2	dfpred2 5689	. 2 |- Pred ( R , A , X ) = ( A i^i { y | y R X } )
4		dfrab2 3903	. 2 |- { y e. A | y R X } = ( { y | y R X } i^i A )
5	1, 3, 4	3eqtr4i 2654	1 |- Pred ( R , A , X ) = { y e. A | y R X }

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   i^i cin 3573   class class class wbr 4653   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-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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  dfpred3g  5691