Theorem bnj548 30967

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
bnj548.1	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj548.2	|- B = U_ y e. ( f ` i ) pred ( y , A , R )
bnj548.3	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj548.4	|- G = ( f u. { <. m , C >. } )
bnj548.5	|- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )

Assertion

Ref	Expression
bnj548	|- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> B = K )

Distinct variable groups:   y, G   y, f   y, i

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

Proof of Theorem bnj548

Step	Hyp	Ref	Expression
1		bnj548.5	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
2	1	bnj930 30840	. . . . . 6 |- ( ( R FrSe A /\ ta /\ si ) -> Fun G )
3	2	adantr 481	. . . . 5 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> Fun G )
4		bnj548.1	. . . . . . . 8 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
5	4	simp1bi 1076	. . . . . . 7 |- ( ta -> f Fn m )
6		fndm 5990	. . . . . . . 8 |- ( f Fn m -> dom f = m )
7		eleq2 2690	. . . . . . . . 9 |- ( dom f = m -> ( i e. dom f <-> i e. m ) )
8	7	biimpar 502	. . . . . . . 8 |- ( ( dom f = m /\ i e. m ) -> i e. dom f )
9	6, 8	sylan 488	. . . . . . 7 |- ( ( f Fn m /\ i e. m ) -> i e. dom f )
10	5, 9	sylan 488	. . . . . 6 |- ( ( ta /\ i e. m ) -> i e. dom f )
11	10	3ad2antl2 1224	. . . . 5 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> i e. dom f )
12	3, 11	jca 554	. . . 4 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> ( Fun G /\ i e. dom f ) )
13		bnj548.4	. . . . 5 |- G = ( f u. { <. m , C >. } )
14	13	bnj931 30841	. . . 4 |- f C_ G
15	12, 14	jctil 560	. . 3 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> ( f C_ G /\ ( Fun G /\ i e. dom f ) ) )
16		3anan12 1051	. . 3 |- ( ( Fun G /\ f C_ G /\ i e. dom f ) <-> ( f C_ G /\ ( Fun G /\ i e. dom f ) ) )
17	15, 16	sylibr 224	. 2 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> ( Fun G /\ f C_ G /\ i e. dom f ) )
18		funssfv 6209	. 2 |- ( ( Fun G /\ f C_ G /\ i e. dom f ) -> ( G ` i ) = ( f ` i ) )
19		iuneq1 4534	. . . 4 |- ( ( G ` i ) = ( f ` i ) -> U_ y e. ( G ` i ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
20	19	eqcomd 2628	. . 3 |- ( ( G ` i ) = ( f ` i ) -> U_ y e. ( f ` i ) pred ( y , A , R ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
21		bnj548.2	. . 3 |- B = U_ y e. ( f ` i ) pred ( y , A , R )
22		bnj548.3	. . 3 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
23	20, 21, 22	3eqtr4g 2681	. 2 |- ( ( G ` i ) = ( f ` i ) -> B = K )
24	17, 18, 23	3syl 18	1 |- ( ( ( R FrSe A /\ ta /\ si ) /\ i e. m ) -> B = K )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   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-rab 2921  df-v 3202  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-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  bnj553  30968