Theorem bnj553 32204

Description: Technical lemma for bnj852 32227. 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
bnj553.1	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj553.2	|- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj553.3	|- D = ( om \ { (/) } )
bnj553.4	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj553.5	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj553.6	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj553.7	|- C = U_ y e. ( f ` p ) pred ( y , A , R )
bnj553.8	|- G = ( f u. { <. m , C >. } )
bnj553.9	|- B = U_ y e. ( f ` i ) pred ( y , A , R )
bnj553.10	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj553.11	|- L = U_ y e. ( G ` p ) pred ( y , A , R )
bnj553.12	|- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )

Assertion

Ref	Expression
bnj553	|- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> ( G ` m ) = L )

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

Allowed substitution hints:   ta( x, y, f, i, m, n, p)   si( x, y, f, i, m, n, p)   A( x, f, m, n)   B( x, y, f, i, m, n, p)   C( x, y, f, i, m, n, p)   D( x, y, f, i, m, n, p)   R( x, f, m, n)   G( x, f, i, m, n, p)   K( x, y, f, i, m, n, p)   L( x, y, f, i, m, n, p)   ph'( x, y, f, i, m, n)   ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj553

Step	Hyp	Ref	Expression
1		bnj553.12	. . . . 5 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
2	1	bnj930 32076	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> Fun G )
3		opex 4659	. . . . . . 7 |- <. m , C >. e. _V
4	3	snid 4008	. . . . . 6 |- <. m , C >. e. { <. m , C >. }
5		elun2 3627	. . . . . 6 |- ( <. m , C >. e. { <. m , C >. } -> <. m , C >. e. ( f u. { <. m , C >. } ) )
6	4, 5	ax-mp 5	. . . . 5 |- <. m , C >. e. ( f u. { <. m , C >. } )
7		bnj553.8	. . . . 5 |- G = ( f u. { <. m , C >. } )
8	6, 7	eleqtrri 2539	. . . 4 |- <. m , C >. e. G
9		funopfv 5835	. . . 4 |- ( Fun G -> ( <. m , C >. e. G -> ( G ` m ) = C ) )
10	2, 8, 9	mpisyl 18	. . 3 |- ( ( R FrSe A /\ ta /\ si ) -> ( G ` m ) = C )
11	10	3ad2ant1 1009	. 2 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> ( G ` m ) = C )
12		fveq2 5794	. . . . . 6 |- ( p = i -> ( G ` p ) = ( G ` i ) )
13	12	bnj1113 32092	. . . . 5 |- ( p = i -> U_ y e. ( G ` p ) pred ( y , A , R ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
14		bnj553.11	. . . . 5 |- L = U_ y e. ( G ` p ) pred ( y , A , R )
15		bnj553.10	. . . . 5 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
16	13, 14, 15	3eqtr4g 2518	. . . 4 |- ( p = i -> L = K )
17	16	3ad2ant3 1011	. . 3 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> L = K )
18		bnj553.5	. . . . 5 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
19		bnj553.9	. . . . 5 |- B = U_ y e. ( f ` i ) pred ( y , A , R )
20		bnj553.4	. . . . 5 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
21	18, 19, 15, 20, 1	bnj548 32203	. . . 4 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> B = K )
22	21	3adant3 1008	. . 3 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> B = K )
23		fveq2 5794	. . . . . 6 |- ( p = i -> ( f ` p ) = ( f ` i ) )
24	23	bnj1113 32092	. . . . 5 |- ( p = i -> U_ y e. ( f ` p ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
25		bnj553.7	. . . . . . 7 |- C = U_ y e. ( f ` p ) pred ( y , A , R )
26	19, 25	eqeq12i 2472	. . . . . 6 |- ( B = C <-> U_ y e. ( f ` i ) pred ( y , A , R ) = U_ y e. ( f ` p ) pred ( y , A , R ) )
27		eqcom 2461	. . . . . 6 |- ( U_ y e. ( f ` i ) pred ( y , A , R ) = U_ y e. ( f ` p ) pred ( y , A , R ) <-> U_ y e. ( f ` p ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
28	26, 27	bitri 249	. . . . 5 |- ( B = C <-> U_ y e. ( f ` p ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
29	24, 28	sylibr 212	. . . 4 |- ( p = i -> B = C )
30	29	3ad2ant3 1011	. . 3 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> B = C )
31	17, 22, 30	3eqtr2rd 2500	. 2 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> C = L )
32	11, 31	eqtrd 2493	1 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> ( G ` m ) = L )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2796   \ cdif 3428   u. cun 3429   (/)c0 3740   {csn 3980   <.cop 3986   U_ciun 4274   suc csuc 4824   Fun wfun 5515   Fn wfn 5516   ` cfv 5521   omcom 6581   predc-bnj14 31989   FrSe w-bnj15 31993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-res 4955  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529

This theorem is referenced by:  bnj557  32207