Theorem bnj938 31007

Description: Technical lemma for bnj69 31078. 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
bnj938.1	|- D = ( om \ { (/) } )
bnj938.2	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj938.3	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj938.4	|- ( ph' <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj938.5	|- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj938	|- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V )

Distinct variable groups:   A, i, p, y   R, i, p, y   f, i, p, y   i, m, p

Allowed substitution hints:   ta( y, f, i, m, n, p)   si( y, f, i, m, n, p)   A( f, m, n)   D( y, f, i, m, n, p)   R( f, m, n)   X( y, f, i, m, n, p)   ph'( y, f, i, m, n, p)   ps'( y, f, i, m, n, p)

Proof of Theorem bnj938

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 3215	. . 3 |- ( X e. A -> E. x x = X )
2	1	bnj706 30824	. 2 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> E. x x = X )
3		bnj291 30777	. . . . . 6 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R FrSe A /\ ta /\ si ) /\ X e. A ) )
4	3	simplbi 476	. . . . 5 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> ( R FrSe A /\ ta /\ si ) )
5		bnj602 30985	. . . . . . . . . 10 |- ( x = X -> pred ( x , A , R ) = pred ( X , A , R ) )
6	5	eqeq2d 2632	. . . . . . . . 9 |- ( x = X -> ( ( f ` (/) ) = pred ( x , A , R ) <-> ( f ` (/) ) = pred ( X , A , R ) ) )
7		bnj938.4	. . . . . . . . 9 |- ( ph' <-> ( f ` (/) ) = pred ( X , A , R ) )
8	6, 7	syl6bbr 278	. . . . . . . 8 |- ( x = X -> ( ( f ` (/) ) = pred ( x , A , R ) <-> ph' ) )
9	8	3anbi2d 1404	. . . . . . 7 |- ( x = X -> ( ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ph' /\ ps' ) ) )
10		bnj938.2	. . . . . . 7 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
11	9, 10	syl6bbr 278	. . . . . 6 |- ( x = X -> ( ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) <-> ta ) )
12	11	3anbi2d 1404	. . . . 5 |- ( x = X -> ( ( R FrSe A /\ ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) /\ si ) <-> ( R FrSe A /\ ta /\ si ) ) )
13	4, 12	syl5ibr 236	. . . 4 |- ( x = X -> ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> ( R FrSe A /\ ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) /\ si ) ) )
14		bnj938.1	. . . . 5 |- D = ( om \ { (/) } )
15		biid 251	. . . . 5 |- ( ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) )
16		bnj938.3	. . . . 5 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
17		biid 251	. . . . 5 |- ( ( f ` (/) ) = pred ( x , A , R ) <-> ( f ` (/) ) = pred ( x , A , R ) )
18		bnj938.5	. . . . 5 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
19	14, 15, 16, 17, 18	bnj546 30966	. . . 4 |- ( ( R FrSe A /\ ( f Fn m /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps' ) /\ si ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V )
20	13, 19	syl6 35	. . 3 |- ( x = X -> ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V ) )
21	20	exlimiv 1858	. 2 |- ( E. x x = X -> ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V ) )
22	2, 21	mpcom 38	1 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` p ) pred ( y , A , R ) e. _V )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  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-om 7066  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj944  31008  bnj969  31016