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Theorem bnj1442 29480
Description: Technical lemma for bnj60 29493. 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 29202 . . . . 5  |-  ( ch 
->  Fun  Q )
5 opex 4429 . . . . . . . 8  |-  <. x ,  ( G `  Z ) >.  e.  _V
65snid 3843 . . . . . . 7  |-  <. x ,  ( G `  Z ) >.  e.  { <. x ,  ( G `
 Z ) >. }
7 elun2 3517 . . . . . . 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 2511 . . . . 5  |-  <. x ,  ( G `  Z ) >.  e.  Q
11 funopfv 5768 . . . . 5  |-  ( Fun 
Q  ->  ( <. x ,  ( G `  Z ) >.  e.  Q  ->  ( Q `  x
)  =  ( G `
 Z ) ) )
124, 10, 11ee10 1386 . . . 4  |-  ( ch 
->  ( Q `  x
)  =  ( G `
 Z ) )
132, 12bnj832 29188 . . 3  |-  ( th 
->  ( Q `  x
)  =  ( G `
 Z ) )
141, 13bnj832 29188 . 2  |-  ( et 
->  ( Q `  x
)  =  ( G `
 Z ) )
15 elsni 3840 . . . 4  |-  ( z  e.  { x }  ->  z  =  x )
161, 15bnj833 29189 . . 3  |-  ( et 
->  z  =  x
)
1716fveq2d 5734 . 2  |-  ( et 
->  ( Q `  z
)  =  ( Q `
 x ) )
18 bnj602 29348 . . . . . . . 8  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1918reseq2d 5148 . . . . . . 7  |-  ( z  =  x  ->  ( Q  |`  pred ( z ,  A ,  R ) )  =  ( Q  |`  pred ( x ,  A ,  R ) ) )
2016, 19syl 16 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( Q  |`  pred (
x ,  A ,  R ) ) )
219bnj931 29203 . . . . . . . . . 10  |-  P  C_  Q
2221a1i 11 . . . . . . . . 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 448 . . . . . . . . . . . 12  |-  ( ps 
->  R  FrSe  A )
2623, 25bnj835 29190 . . . . . . . . . . 11  |-  ( ch 
->  R  FrSe  A )
27 bnj1442.5 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  -.  E. f ta }
2827, 23bnj1212 29233 . . . . . . . . . . 11  |-  ( ch 
->  x  e.  A
)
29 bnj906 29363 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3026, 28, 29syl2anc 644 . . . . . . . . . 10  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
31 bnj1442.15 . . . . . . . . . . 11  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
32 fndm 5546 . . . . . . . . . . 11  |-  ( P  Fn  trCl ( x ,  A ,  R )  ->  dom  P  =  trCl ( x ,  A ,  R ) )
3331, 32syl 16 . . . . . . . . . 10  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
3430, 33sseqtr4d 3387 . . . . . . . . 9  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  P )
354, 22, 34bnj1503 29282 . . . . . . . 8  |-  ( ch 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
362, 35bnj832 29188 . . . . . . 7  |-  ( th 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
371, 36bnj832 29188 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3820, 37eqtrd 2470 . . . . 5  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3916, 38opeq12d 3994 . . . 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 2495 . . 3  |-  ( et 
->  W  =  Z
)
4342fveq2d 5734 . 2  |-  ( et 
->  ( G `  W
)  =  ( G `
 Z ) )
4414, 17, 433eqtr4d 2480 1  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   [.wsbc 3163    u. cun 3320    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U.cuni 4017   class class class wbr 4214   dom cdm 4880    |` cres 4882   Fun wfun 5450    Fn wfn 5451   ` cfv 5456    predc-bnj14 29114    FrSe w-bnj15 29118    trClc-bnj18 29120
This theorem is referenced by:  bnj1423  29482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-bnj17 29113  df-bnj14 29115  df-bnj13 29117  df-bnj15 29119  df-bnj18 29121
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