Theorem bnj553 34303

Description: Technical lemma for bnj852 34326. 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 34175	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> Fun G )
3		opex 4626	. . . . . . 7 |- <. m , C >. e. _V
4	3	snid 3972	. . . . . 6 |- <. m , C >. e. { <. m , C >. }
5		elun2 3586	. . . . . 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 2469	. . . 4 |- <. m , C >. e. G
9		funopfv 5813	. . . 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 1015	. 2 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> ( G ` m ) = C )
12		fveq2 5774	. . . . . 6 |- ( p = i -> ( G ` p ) = ( G ` i ) )
13	12	bnj1113 34191	. . . . 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 2448	. . . 4 |- ( p = i -> L = K )
17	16	3ad2ant3 1017	. . 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 34302	. . . 4 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> B = K )
22	21	3adant3 1014	. . 3 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> B = K )
23		fveq2 5774	. . . . . 6 |- ( p = i -> ( f ` p ) = ( f ` i ) )
24	23	bnj1113 34191	. . . . 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 2402	. . . . . 6 |- ( B = C <-> U_ y e. ( f ` i ) pred ( y , A , R ) = U_ y e. ( f ` p ) pred ( y , A , R ) )
27		eqcom 2391	. . . . . 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 1017	. . 3 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> B = C )
31	17, 22, 30	3eqtr2rd 2430	. 2 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> C = L )
32	11, 31	eqtrd 2423	1 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> ( G ` m ) = L )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 971   = wceq 1399   e. wcel 1826   A.wral 2732   \ cdif 3386   u. cun 3387   (/)c0 3711   {csn 3944   <.cop 3950   U_ciun 4243   suc csuc 4794   Fun wfun 5490   Fn wfn 5491   ` cfv 5496   omcom 6599   predc-bnj14 34087   FrSe w-bnj15 34091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504

This theorem is referenced by:  bnj557  34306