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Theorem bnj1442 29395
Description: Technical lemma for bnj60 29408. 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
bnj1442.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1442.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1442.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1442.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1442.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1442.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1442.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1442.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1442.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1442.10  |-  P  = 
U. H
bnj1442.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1442.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1442.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1442.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1442.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1442.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1442.17  |-  ( th  <->  ( ch  /\  z  e.  E ) )
bnj1442.18  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
Assertion
Ref Expression
bnj1442  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Distinct variable group:    x, A
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, z, f, d)    th( x, y, z, f, d)    ta( x, y, z, f, d)    et( x, y, z, f, d)    A( y, z, f, d)    B( x, y, z, f, d)    C( x, y, z, f, d)    D( x, y, z, f, d)    P( x, y, z, f, d)    Q( x, y, z, f, d)    R( x, y, z, f, d)    E( x, y, z, f, d)    G( x, y, z, f, d)    H( x, y, z, f, d)    W( x, y, z, f, d)    Y( x, y, z, f, d)    Z( x, y, z, f, d)    ta'( x, y, z, f, d)

Proof of Theorem bnj1442
StepHypRef Expression
1 bnj1442.18 . . 3  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
2 bnj1442.17 . . . 4  |-  ( th  <->  ( ch  /\  z  e.  E ) )
3 bnj1442.16 . . . . . 6  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
43bnj930 29117 . . . . 5  |-  ( ch 
->  Fun  Q )
5 opex 4253 . . . . . . . 8  |-  <. x ,  ( G `  Z ) >.  e.  _V
65snid 3680 . . . . . . 7  |-  <. x ,  ( G `  Z ) >.  e.  { <. x ,  ( G `
 Z ) >. }
7 elun2 3356 . . . . . . 7  |-  ( <.
x ,  ( G `
 Z ) >.  e.  { <. x ,  ( G `  Z )
>. }  ->  <. x ,  ( G `  Z
) >.  e.  ( P  u.  { <. x ,  ( G `  Z ) >. } ) )
86, 7ax-mp 8 . . . . . 6  |-  <. x ,  ( G `  Z ) >.  e.  ( P  u.  { <. x ,  ( G `  Z ) >. } )
9 bnj1442.12 . . . . . 6  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
108, 9eleqtrri 2369 . . . . 5  |-  <. x ,  ( G `  Z ) >.  e.  Q
11 funopfv 5578 . . . . 5  |-  ( Fun 
Q  ->  ( <. x ,  ( G `  Z ) >.  e.  Q  ->  ( Q `  x
)  =  ( G `
 Z ) ) )
124, 10, 11ee10 1366 . . . 4  |-  ( ch 
->  ( Q `  x
)  =  ( G `
 Z ) )
132, 12bnj832 29103 . . 3  |-  ( th 
->  ( Q `  x
)  =  ( G `
 Z ) )
141, 13bnj832 29103 . 2  |-  ( et 
->  ( Q `  x
)  =  ( G `
 Z ) )
15 elsni 3677 . . . 4  |-  ( z  e.  { x }  ->  z  =  x )
161, 15bnj833 29104 . . 3  |-  ( et 
->  z  =  x
)
1716fveq2d 5545 . 2  |-  ( et 
->  ( Q `  z
)  =  ( Q `
 x ) )
18 bnj602 29263 . . . . . . . 8  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1918reseq2d 4971 . . . . . . 7  |-  ( z  =  x  ->  ( Q  |`  pred ( z ,  A ,  R ) )  =  ( Q  |`  pred ( x ,  A ,  R ) ) )
2016, 19syl 15 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( Q  |`  pred (
x ,  A ,  R ) ) )
219bnj931 29118 . . . . . . . . . 10  |-  P  C_  Q
2221a1i 10 . . . . . . . . 9  |-  ( ch 
->  P  C_  Q )
23 bnj1442.7 . . . . . . . . . . . 12  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
24 bnj1442.6 . . . . . . . . . . . . 13  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
2524simplbi 446 . . . . . . . . . . . 12  |-  ( ps 
->  R  FrSe  A )
2623, 25bnj835 29105 . . . . . . . . . . 11  |-  ( ch 
->  R  FrSe  A )
27 bnj1442.5 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  -.  E. f ta }
2827, 23bnj1212 29148 . . . . . . . . . . 11  |-  ( ch 
->  x  e.  A
)
29 bnj906 29278 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3026, 28, 29syl2anc 642 . . . . . . . . . 10  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
31 bnj1442.15 . . . . . . . . . . 11  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
32 fndm 5359 . . . . . . . . . . 11  |-  ( P  Fn  trCl ( x ,  A ,  R )  ->  dom  P  =  trCl ( x ,  A ,  R ) )
3331, 32syl 15 . . . . . . . . . 10  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
3430, 33sseqtr4d 3228 . . . . . . . . 9  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  P )
354, 22, 34bnj1503 29197 . . . . . . . 8  |-  ( ch 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
362, 35bnj832 29103 . . . . . . 7  |-  ( th 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
371, 36bnj832 29103 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3820, 37eqtrd 2328 . . . . 5  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3916, 38opeq12d 3820 . . . 4  |-  ( et 
->  <. z ,  ( Q  |`  pred ( z ,  A ,  R
) ) >.  =  <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
)
40 bnj1442.13 . . . 4  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
41 bnj1442.11 . . . 4  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
4239, 40, 413eqtr4g 2353 . . 3  |-  ( et 
->  W  =  Z
)
4342fveq2d 5545 . 2  |-  ( et 
->  ( G `  W
)  =  ( G `
 Z ) )
4414, 17, 433eqtr4d 2338 1  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   [.wsbc 3004    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U.cuni 3843   class class class wbr 4039   dom cdm 4705    |` cres 4707   Fun wfun 5265    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1423  29397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036
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