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Theorem bnj1501 34543
Description: Technical lemma for bnj1500 34544. 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
bnj1501.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1501.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1501.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1501.4  |-  F  = 
U. C
bnj1501.5  |-  ( ph  <->  ( R  FrSe  A  /\  x  e.  A )
)
bnj1501.6  |-  ( ps  <->  (
ph  /\  f  e.  C  /\  x  e.  dom  f ) )
bnj1501.7  |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )
Assertion
Ref Expression
bnj1501  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    R, d, f, x    Y, d    ph, d, f
Allowed substitution hints:    ph( x)    ps( x, f, d)    ch( x, f, d)    B( x, d)    C( x, f, d)    F( x, f, d)    Y( x, f)

Proof of Theorem bnj1501
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1501.5 . 2  |-  ( ph  <->  ( R  FrSe  A  /\  x  e.  A )
)
21simprbi 462 . . . . . . . 8  |-  ( ph  ->  x  e.  A )
3 bnj1501.1 . . . . . . . . . . 11  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
4 bnj1501.2 . . . . . . . . . . 11  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
5 bnj1501.3 . . . . . . . . . . 11  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
6 bnj1501.4 . . . . . . . . . . 11  |-  F  = 
U. C
73, 4, 5, 6bnj60 34538 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  F  Fn  A )
8 fndm 5662 . . . . . . . . . 10  |-  ( F  Fn  A  ->  dom  F  =  A )
97, 8syl 16 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  dom  F  =  A )
101, 9bnj832 34235 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
112, 10eleqtrrd 2545 . . . . . . 7  |-  ( ph  ->  x  e.  dom  F
)
126dmeqi 5193 . . . . . . . 8  |-  dom  F  =  dom  U. C
135bnj1317 34300 . . . . . . . . 9  |-  ( w  e.  C  ->  A. f  w  e.  C )
1413bnj1400 34314 . . . . . . . 8  |-  dom  U. C  =  U_ f  e.  C  dom  f
1512, 14eqtri 2483 . . . . . . 7  |-  dom  F  =  U_ f  e.  C  dom  f
1611, 15syl6eleq 2552 . . . . . 6  |-  ( ph  ->  x  e.  U_ f  e.  C  dom  f )
1716bnj1405 34315 . . . . 5  |-  ( ph  ->  E. f  e.  C  x  e.  dom  f )
18 bnj1501.6 . . . . 5  |-  ( ps  <->  (
ph  /\  f  e.  C  /\  x  e.  dom  f ) )
1917, 18bnj1209 34275 . . . 4  |-  ( ph  ->  E. f ps )
205bnj1436 34318 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
2120bnj1299 34297 . . . . . . . . 9  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
22 fndm 5662 . . . . . . . . 9  |-  ( f  Fn  d  ->  dom  f  =  d )
2321, 22bnj31 34192 . . . . . . . 8  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
2418, 23bnj836 34238 . . . . . . 7  |-  ( ps 
->  E. d  e.  B  dom  f  =  d
)
25 bnj1501.7 . . . . . . 7  |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )
263, 4, 5, 6, 1, 18bnj1518 34540 . . . . . . 7  |-  ( ps 
->  A. d ps )
2724, 25, 26bnj1521 34329 . . . . . 6  |-  ( ps 
->  E. d ch )
287bnj930 34248 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  Fun  F
)
291, 28bnj832 34235 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
3018, 29bnj835 34237 . . . . . . . . . 10  |-  ( ps 
->  Fun  F )
31 elssuni 4264 . . . . . . . . . . . 12  |-  ( f  e.  C  ->  f  C_ 
U. C )
3231, 6syl6sseqr 3536 . . . . . . . . . . 11  |-  ( f  e.  C  ->  f  C_  F )
3318, 32bnj836 34238 . . . . . . . . . 10  |-  ( ps 
->  f  C_  F )
3418simp3bi 1011 . . . . . . . . . 10  |-  ( ps 
->  x  e.  dom  f )
3530, 33, 34bnj1502 34326 . . . . . . . . 9  |-  ( ps 
->  ( F `  x
)  =  ( f `
 x ) )
363, 4, 5bnj1514 34539 . . . . . . . . . . 11  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
3718, 36bnj836 34238 . . . . . . . . . 10  |-  ( ps 
->  A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
3837, 34bnj1294 34296 . . . . . . . . 9  |-  ( ps 
->  ( f `  x
)  =  ( G `
 Y ) )
3935, 38eqtrd 2495 . . . . . . . 8  |-  ( ps 
->  ( F `  x
)  =  ( G `
 Y ) )
4025, 39bnj835 34237 . . . . . . 7  |-  ( ch 
->  ( F `  x
)  =  ( G `
 Y ) )
4125, 30bnj835 34237 . . . . . . . . . . 11  |-  ( ch 
->  Fun  F )
4225, 33bnj835 34237 . . . . . . . . . . 11  |-  ( ch 
->  f  C_  F )
433bnj1517 34328 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
4425, 43bnj836 34238 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
4525, 34bnj835 34237 . . . . . . . . . . . . . 14  |-  ( ch 
->  x  e.  dom  f )
4625simp3bi 1011 . . . . . . . . . . . . . 14  |-  ( ch 
->  dom  f  =  d )
4745, 46eleqtrd 2544 . . . . . . . . . . . . 13  |-  ( ch 
->  x  e.  d
)
4844, 47bnj1294 34296 . . . . . . . . . . . 12  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  d )
4948, 46sseqtr4d 3526 . . . . . . . . . . 11  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  f )
5041, 42, 49bnj1503 34327 . . . . . . . . . 10  |-  ( ch 
->  ( F  |`  pred (
x ,  A ,  R ) )  =  ( f  |`  pred (
x ,  A ,  R ) ) )
5150opeq2d 4210 . . . . . . . . 9  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
)
5251, 4syl6eqr 2513 . . . . . . . 8  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  Y )
5352fveq2d 5852 . . . . . . 7  |-  ( ch 
->  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
5440, 53eqtr4d 2498 . . . . . 6  |-  ( ch 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5527, 54bnj593 34222 . . . . 5  |-  ( ps 
->  E. d ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
563, 4, 5, 6bnj1519 34541 . . . . 5  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5755, 56bnj1397 34313 . . . 4  |-  ( ps 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5819, 57bnj593 34222 . . 3  |-  ( ph  ->  E. f ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
593, 4, 5, 6bnj1520 34542 . . 3  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
6058, 59bnj1397 34313 . 2  |-  ( ph  ->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
611, 60bnj1459 34321 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805    C_ wss 3461   <.cop 4022   U.cuni 4235   U_ciun 4315   dom cdm 4988    |` cres 4990   Fun wfun 5564    Fn wfn 5565   ` cfv 5570    predc-bnj14 34160    FrSe w-bnj15 34164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-bnj17 34159  df-bnj14 34161  df-bnj13 34163  df-bnj15 34165  df-bnj18 34167  df-bnj19 34169
This theorem is referenced by:  bnj1500  34544
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