Theorem bnj149 30945

Description: Technical lemma for bnj151 30947. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj149.1	|- ( th1 <-> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) ) )
bnj149.2	|- ( ze0 <-> ( f Fn 1o /\ ph' /\ ps' ) )
bnj149.3	|- ( ze1 <-> [. g / f ]. ze0 )
bnj149.4	|- ( ph1 <-> [. g / f ]. ph' )
bnj149.5	|- ( ps1 <-> [. g / f ]. ps' )
bnj149.6	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )

Assertion

Ref	Expression
bnj149	|- th1

Distinct variable groups:   A, f, g, x   R, f, g, x   f, ze1   g, ze0

Allowed substitution hints:   ph'( x, f, g)   ps'( x, f, g)   ze0( x, f)   ph1( x, f, g)   ps1( x, f, g)   th1( x, f, g)   ze1( x, g)

Proof of Theorem bnj149

Step	Hyp	Ref	Expression
1		simpr1 1067	. . . . . . . 8 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> f Fn 1o )
2		df1o2 7572	. . . . . . . . 9 |- 1o = { (/) }
3	2	fneq2i 5986	. . . . . . . 8 |- ( f Fn 1o <-> f Fn { (/) } )
4	1, 3	sylib 208	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> f Fn { (/) } )
5		simpr2 1068	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> ph' )
6		bnj149.6	. . . . . . . . . 10 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
7	5, 6	sylib 208	. . . . . . . . 9 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> ( f ` (/) ) = pred ( x , A , R ) )
8		fvex 6201	. . . . . . . . . 10 |- ( f ` (/) ) e. _V
9	8	elsn 4192	. . . . . . . . 9 |- ( ( f ` (/) ) e. { pred ( x , A , R ) } <-> ( f ` (/) ) = pred ( x , A , R ) )
10	7, 9	sylibr 224	. . . . . . . 8 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> ( f ` (/) ) e. { pred ( x , A , R ) } )
11		0ex 4790	. . . . . . . . 9 |- (/) e. _V
12		fveq2 6191	. . . . . . . . . 10 |- ( g = (/) -> ( f ` g ) = ( f ` (/) ) )
13	12	eleq1d 2686	. . . . . . . . 9 |- ( g = (/) -> ( ( f ` g ) e. { pred ( x , A , R ) } <-> ( f ` (/) ) e. { pred ( x , A , R ) } ) )
14	11, 13	ralsn 4222	. . . . . . . 8 |- ( A. g e. { (/) } ( f ` g ) e. { pred ( x , A , R ) } <-> ( f ` (/) ) e. { pred ( x , A , R ) } )
15	10, 14	sylibr 224	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> A. g e. { (/) } ( f ` g ) e. { pred ( x , A , R ) } )
16		ffnfv 6388	. . . . . . 7 |- ( f : { (/) } --> { pred ( x , A , R ) } <-> ( f Fn { (/) } /\ A. g e. { (/) } ( f ` g ) e. { pred ( x , A , R ) } ) )
17	4, 15, 16	sylanbrc 698	. . . . . 6 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> f : { (/) } --> { pred ( x , A , R ) } )
18		bnj93 30933	. . . . . . . 8 |- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) e. _V )
19	18	adantr 481	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> pred ( x , A , R ) e. _V )
20		fsng 6404	. . . . . . 7 |- ( ( (/) e. _V /\ pred ( x , A , R ) e. _V ) -> ( f : { (/) } --> { pred ( x , A , R ) } <-> f = { <. (/) , pred ( x , A , R ) >. } ) )
21	11, 19, 20	sylancr 695	. . . . . 6 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> ( f : { (/) } --> { pred ( x , A , R ) } <-> f = { <. (/) , pred ( x , A , R ) >. } ) )
22	17, 21	mpbid 222	. . . . 5 |- ( ( ( R FrSe A /\ x e. A ) /\ ( f Fn 1o /\ ph' /\ ps' ) ) -> f = { <. (/) , pred ( x , A , R ) >. } )
23	22	ex 450	. . . 4 |- ( ( R FrSe A /\ x e. A ) -> ( ( f Fn 1o /\ ph' /\ ps' ) -> f = { <. (/) , pred ( x , A , R ) >. } ) )
24	23	alrimiv 1855	. . 3 |- ( ( R FrSe A /\ x e. A ) -> A. f ( ( f Fn 1o /\ ph' /\ ps' ) -> f = { <. (/) , pred ( x , A , R ) >. } ) )
25		mo2icl 3385	. . 3 |- ( A. f ( ( f Fn 1o /\ ph' /\ ps' ) -> f = { <. (/) , pred ( x , A , R ) >. } ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) )
26	24, 25	syl 17	. 2 |- ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) )
27		bnj149.1	. 2 |- ( th1 <-> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) ) )
28	26, 27	mpbir 221	1 |- th1

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   A.wral 2912   _Vcvv 3200   [.wsbc 3435   (/)c0 3915   {csn 4177   <.cop 4183   Fn wfn 5883   -->wf 5884   ` cfv 5888   1oc1o 7553   predc-bnj14 30754   FrSe w-bnj15 30758

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-reu 2919  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj151  30947