Theorem setlikespec 5701

Description: If R is set-like in A, then all predecessors classes of elements of A exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
setlikespec	|- ( ( X e. A /\ R Se A ) -> Pred ( R , A , X ) e. _V )

Proof of Theorem setlikespec

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2	1	elpred 5693	. . . . 5 |- ( X e. A -> ( x e. Pred ( R , A , X ) <-> ( x e. A /\ x R X ) ) )
3	2	abbi2dv 2742	. . . 4 |- ( X e. A -> Pred ( R , A , X ) = { x | ( x e. A /\ x R X ) } )
4		df-rab 2921	. . . 4 |- { x e. A | x R X } = { x | ( x e. A /\ x R X ) }
5	3, 4	syl6reqr 2675	. . 3 |- ( X e. A -> { x e. A | x R X } = Pred ( R , A , X ) )
6	5	adantr 481	. 2 |- ( ( X e. A /\ R Se A ) -> { x e. A | x R X } = Pred ( R , A , X ) )
7		seex 5077	. . 3 |- ( ( R Se A /\ X e. A ) -> { x e. A | x R X } e. _V )
8	7	ancoms 469	. 2 |- ( ( X e. A /\ R Se A ) -> { x e. A | x R X } e. _V )
9	6, 8	eqeltrrd 2702	1 |- ( ( X e. A /\ R Se A ) -> Pred ( R , A , X ) e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   class class class wbr 4653   Se wse 5071   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-se 5074  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:  wfrlem15  7429  trpredtr  31730  trpredmintr  31731  trpredelss  31732  dftrpred3g  31733  trpredpo  31735  trpredrec  31738  frmin  31739