Theorem bnj545 30965

Description: Technical lemma for bnj852 30991. 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.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj545.1	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj545.2	|- D = ( om \ { (/) } )
bnj545.3	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj545.4	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj545.5	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj545.6	|- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
bnj545.7	|- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )

Assertion

Ref	Expression
bnj545	|- ( ( R FrSe A /\ ta /\ si ) -> ph" )

Proof of Theorem bnj545

Step	Hyp	Ref	Expression
1		bnj545.4	. . . . . . . . . 10 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
2	1	simp1bi 1076	. . . . . . . . 9 |- ( ta -> f Fn m )
3		bnj545.5	. . . . . . . . . 10 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
4	3	simp1bi 1076	. . . . . . . . 9 |- ( si -> m e. D )
5	2, 4	anim12i 590	. . . . . . . 8 |- ( ( ta /\ si ) -> ( f Fn m /\ m e. D ) )
6	5	3adant1 1079	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> ( f Fn m /\ m e. D ) )
7		bnj545.2	. . . . . . . . 9 |- D = ( om \ { (/) } )
8	7	bnj529 30811	. . . . . . . 8 |- ( m e. D -> (/) e. m )
9		fndm 5990	. . . . . . . 8 |- ( f Fn m -> dom f = m )
10		eleq2 2690	. . . . . . . . 9 |- ( dom f = m -> ( (/) e. dom f <-> (/) e. m ) )
11	10	biimparc 504	. . . . . . . 8 |- ( ( (/) e. m /\ dom f = m ) -> (/) e. dom f )
12	8, 9, 11	syl2anr 495	. . . . . . 7 |- ( ( f Fn m /\ m e. D ) -> (/) e. dom f )
13	6, 12	syl 17	. . . . . 6 |- ( ( R FrSe A /\ ta /\ si ) -> (/) e. dom f )
14		bnj545.6	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
15	14	bnj930 30840	. . . . . 6 |- ( ( R FrSe A /\ ta /\ si ) -> Fun G )
16	13, 15	jca 554	. . . . 5 |- ( ( R FrSe A /\ ta /\ si ) -> ( (/) e. dom f /\ Fun G ) )
17		bnj545.3	. . . . . 6 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
18	17	bnj931 30841	. . . . 5 |- f C_ G
19	16, 18	jctil 560	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> ( f C_ G /\ ( (/) e. dom f /\ Fun G ) ) )
20		df-3an 1039	. . . . 5 |- ( ( (/) e. dom f /\ Fun G /\ f C_ G ) <-> ( ( (/) e. dom f /\ Fun G ) /\ f C_ G ) )
21		3anrot 1043	. . . . 5 |- ( ( (/) e. dom f /\ Fun G /\ f C_ G ) <-> ( Fun G /\ f C_ G /\ (/) e. dom f ) )
22		ancom 466	. . . . 5 |- ( ( ( (/) e. dom f /\ Fun G ) /\ f C_ G ) <-> ( f C_ G /\ ( (/) e. dom f /\ Fun G ) ) )
23	20, 21, 22	3bitr3i 290	. . . 4 |- ( ( Fun G /\ f C_ G /\ (/) e. dom f ) <-> ( f C_ G /\ ( (/) e. dom f /\ Fun G ) ) )
24	19, 23	sylibr 224	. . 3 |- ( ( R FrSe A /\ ta /\ si ) -> ( Fun G /\ f C_ G /\ (/) e. dom f ) )
25		funssfv 6209	. . 3 |- ( ( Fun G /\ f C_ G /\ (/) e. dom f ) -> ( G ` (/) ) = ( f ` (/) ) )
26	24, 25	syl 17	. 2 |- ( ( R FrSe A /\ ta /\ si ) -> ( G ` (/) ) = ( f ` (/) ) )
27	1	simp2bi 1077	. . 3 |- ( ta -> ph' )
28	27	3ad2ant2 1083	. 2 |- ( ( R FrSe A /\ ta /\ si ) -> ph' )
29		bnj545.1	. . . 4 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
30		eqtr 2641	. . . 4 |- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ( f ` (/) ) = pred ( x , A , R ) ) -> ( G ` (/) ) = pred ( x , A , R ) )
31	29, 30	sylan2b 492	. . 3 |- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ph' ) -> ( G ` (/) ) = pred ( x , A , R ) )
32		bnj545.7	. . 3 |- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
33	31, 32	sylibr 224	. 2 |- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ph' ) -> ph" )
34	26, 28, 33	syl2anc 693	1 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   omcom 7065   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-8 1992  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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066

This theorem is referenced by:  bnj600  30989  bnj908  31001