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Theorem axdclem 8146
Description: Lemma for axdc 8148. (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdclem.1  |-  F  =  ( rec ( ( y  e.  _V  |->  ( g `  { z  |  y x z } ) ) ,  s )  |`  om )
Assertion
Ref Expression
axdclem  |-  ( ( A. y  e.  ~P  dom  x ( y  =/=  (/)  ->  ( g `  y )  e.  y )  /\  ran  x  C_ 
dom  x  /\  E. z ( F `  K ) x z )  ->  ( K  e.  om  ->  ( F `  K ) x ( F `  suc  K
) ) )
Distinct variable groups:    y, F, z    y, K, z    y,
g    y, s    x, y, z
Allowed substitution hints:    F( x, g, s)    K( x, g, s)

Proof of Theorem axdclem
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . . . . . . . 9  |-  ( F `
 K )  e. 
_V
2 vex 2791 . . . . . . . . 9  |-  z  e. 
_V
31, 2brelrn 4909 . . . . . . . 8  |-  ( ( F `  K ) x z  ->  z  e.  ran  x )
43abssi 3248 . . . . . . 7  |-  { z  |  ( F `  K ) x z }  C_  ran  x
5 sstr 3187 . . . . . . 7  |-  ( ( { z  |  ( F `  K ) x z }  C_  ran  x  /\  ran  x  C_ 
dom  x )  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
64, 5mpan 651 . . . . . 6  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
7 vex 2791 . . . . . . . 8  |-  x  e. 
_V
87dmex 4941 . . . . . . 7  |-  dom  x  e.  _V
98elpw2 4175 . . . . . 6  |-  ( { z  |  ( F `
 K ) x z }  e.  ~P dom  x  <->  { z  |  ( F `  K ) x z }  C_  dom  x )
106, 9sylibr 203 . . . . 5  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  e.  ~P dom  x )
11 neeq1 2454 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  { z  |  ( F `  K ) x z }  =/=  (/) ) )
12 abn0 3473 . . . . . . . 8  |-  ( { z  |  ( F `
 K ) x z }  =/=  (/)  <->  E. z
( F `  K
) x z )
1311, 12syl6bb 252 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  E. z ( F `
 K ) x z ) )
14 eleq2 2344 . . . . . . . . . 10  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
z  |  ( F `
 K ) x z } ) )
15 breq2 4027 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
( F `  K
) x w  <->  ( F `  K ) x z ) )
1615cbvabv 2402 . . . . . . . . . . 11  |-  { w  |  ( F `  K ) x w }  =  { z  |  ( F `  K ) x z }
1716eleq2i 2347 . . . . . . . . . 10  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( g `  y )  e.  { z  |  ( F `  K
) x z } )
1814, 17syl6bbr 254 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
w  |  ( F `
 K ) x w } ) )
19 fvex 5539 . . . . . . . . . 10  |-  ( g `
 y )  e. 
_V
20 breq2 4027 . . . . . . . . . 10  |-  ( w  =  ( g `  y )  ->  (
( F `  K
) x w  <->  ( F `  K ) x ( g `  y ) ) )
2119, 20elab 2914 . . . . . . . . 9  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( F `  K ) x ( g `  y ) )
2218, 21syl6bb 252 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  y ) ) )
23 fveq2 5525 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( g `  y )  =  ( g `  { z  |  ( F `  K ) x z } ) )
2423breq2d 4035 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( F `
 K ) x ( g `  y
)  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2522, 24bitrd 244 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2613, 25imbi12d 311 . . . . . 6  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( y  =/=  (/)  ->  ( g `  y )  e.  y )  <->  ( E. z
( F `  K
) x z  -> 
( F `  K
) x ( g `
 { z  |  ( F `  K
) x z } ) ) ) )
2726rspcv 2880 . . . . 5  |-  ( { z  |  ( F `
 K ) x z }  e.  ~P dom  x  ->  ( A. y  e.  ~P  dom  x
( y  =/=  (/)  ->  (
g `  y )  e.  y )  ->  ( E. z ( F `  K ) x z  ->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) ) )
2810, 27syl 15 . . . 4  |-  ( ran  x  C_  dom  x  -> 
( A. y  e. 
~P  dom  x (
y  =/=  (/)  ->  (
g `  y )  e.  y )  ->  ( E. z ( F `  K ) x z  ->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) ) )
2928com12 27 . . 3  |-  ( A. y  e.  ~P  dom  x
( y  =/=  (/)  ->  (
g `  y )  e.  y )  ->  ( ran  x  C_  dom  x  -> 
( E. z ( F `  K ) x z  ->  ( F `  K )
x ( g `  { z  |  ( F `  K ) x z } ) ) ) )
30293imp 1145 . 2  |-  ( ( A. y  e.  ~P  dom  x ( y  =/=  (/)  ->  ( g `  y )  e.  y )  /\  ran  x  C_ 
dom  x  /\  E. z ( F `  K ) x z )  ->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) )
31 fvex 5539 . . . 4  |-  ( g `
 { z  |  ( F `  K
) x z } )  e.  _V
32 nfcv 2419 . . . . 5  |-  F/_ y
s
33 nfcv 2419 . . . . 5  |-  F/_ y K
34 nfcv 2419 . . . . . 6  |-  F/_ y
g
35 axdclem.1 . . . . . . . . . 10  |-  F  =  ( rec ( ( y  e.  _V  |->  ( g `  { z  |  y x z } ) ) ,  s )  |`  om )
36 nfmpt1 4109 . . . . . . . . . . . 12  |-  F/_ y
( y  e.  _V  |->  ( g `  {
z  |  y x z } ) )
3736, 32nfrdg 6427 . . . . . . . . . . 11  |-  F/_ y rec ( ( y  e. 
_V  |->  ( g `  { z  |  y x z } ) ) ,  s )
38 nfcv 2419 . . . . . . . . . . 11  |-  F/_ y om
3937, 38nfres 4957 . . . . . . . . . 10  |-  F/_ y
( rec ( ( y  e.  _V  |->  ( g `  { z  |  y x z } ) ) ,  s )  |`  om )
4035, 39nfcxfr 2416 . . . . . . . . 9  |-  F/_ y F
4140, 33nffv 5532 . . . . . . . 8  |-  F/_ y
( F `  K
)
42 nfcv 2419 . . . . . . . 8  |-  F/_ y
x
43 nfcv 2419 . . . . . . . 8  |-  F/_ y
z
4441, 42, 43nfbr 4067 . . . . . . 7  |-  F/ y ( F `  K
) x z
4544nfab 2423 . . . . . 6  |-  F/_ y { z  |  ( F `  K ) x z }
4634, 45nffv 5532 . . . . 5  |-  F/_ y
( g `  {
z  |  ( F `
 K ) x z } )
47 breq1 4026 . . . . . . 7  |-  ( y  =  ( F `  K )  ->  (
y x z  <->  ( F `  K ) x z ) )
4847abbidv 2397 . . . . . 6  |-  ( y  =  ( F `  K )  ->  { z  |  y x z }  =  { z  |  ( F `  K ) x z } )
4948fveq2d 5529 . . . . 5  |-  ( y  =  ( F `  K )  ->  (
g `  { z  |  y x z } )  =  ( g `  { z  |  ( F `  K ) x z } ) )
5032, 33, 46, 35, 49frsucmpt 6450 . . . 4  |-  ( ( K  e.  om  /\  ( g `  {
z  |  ( F `
 K ) x z } )  e. 
_V )  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
5131, 50mpan2 652 . . 3  |-  ( K  e.  om  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
5251breq2d 4035 . 2  |-  ( K  e.  om  ->  (
( F `  K
) x ( F `
 suc  K )  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
5330, 52syl5ibrcom 213 1  |-  ( ( A. y  e.  ~P  dom  x ( y  =/=  (/)  ->  ( g `  y )  e.  y )  /\  ran  x  C_ 
dom  x  /\  E. z ( F `  K ) x z )  ->  ( K  e.  om  ->  ( F `  K ) x ( F `  suc  K
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   suc csuc 4394   omcom 4656   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  axdclem2  8147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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