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Theorem axdclem 8947
Description: Lemma for axdc 8949. (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 5891 . . . . . . . . 9  |-  ( F `
 K )  e. 
_V
2 vex 3090 . . . . . . . . 9  |-  z  e. 
_V
31, 2brelrn 5085 . . . . . . . 8  |-  ( ( F `  K ) x z  ->  z  e.  ran  x )
43abssi 3542 . . . . . . 7  |-  { z  |  ( F `  K ) x z }  C_  ran  x
5 sstr 3478 . . . . . . 7  |-  ( ( { z  |  ( F `  K ) x z }  C_  ran  x  /\  ran  x  C_ 
dom  x )  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
64, 5mpan 674 . . . . . 6  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
7 vex 3090 . . . . . . . 8  |-  x  e. 
_V
87dmex 6740 . . . . . . 7  |-  dom  x  e.  _V
98elpw2 4589 . . . . . 6  |-  ( { z  |  ( F `
 K ) x z }  e.  ~P dom  x  <->  { z  |  ( F `  K ) x z }  C_  dom  x )
106, 9sylibr 215 . . . . 5  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  e.  ~P dom  x )
11 neeq1 2712 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  { z  |  ( F `  K ) x z }  =/=  (/) ) )
12 abn0 3787 . . . . . . . 8  |-  ( { z  |  ( F `
 K ) x z }  =/=  (/)  <->  E. z
( F `  K
) x z )
1311, 12syl6bb 264 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  E. z ( F `
 K ) x z ) )
14 eleq2 2502 . . . . . . . . . 10  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
z  |  ( F `
 K ) x z } ) )
15 breq2 4430 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
( F `  K
) x w  <->  ( F `  K ) x z ) )
1615cbvabv 2572 . . . . . . . . . . 11  |-  { w  |  ( F `  K ) x w }  =  { z  |  ( F `  K ) x z }
1716eleq2i 2507 . . . . . . . . . 10  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( g `  y )  e.  { z  |  ( F `  K
) x z } )
1814, 17syl6bbr 266 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
w  |  ( F `
 K ) x w } ) )
19 fvex 5891 . . . . . . . . . 10  |-  ( g `
 y )  e. 
_V
20 breq2 4430 . . . . . . . . . 10  |-  ( w  =  ( g `  y )  ->  (
( F `  K
) x w  <->  ( F `  K ) x ( g `  y ) ) )
2119, 20elab 3224 . . . . . . . . 9  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( F `  K ) x ( g `  y ) )
2218, 21syl6bb 264 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  y ) ) )
23 fveq2 5881 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( g `  y )  =  ( g `  { z  |  ( F `  K ) x z } ) )
2423breq2d 4438 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( F `
 K ) x ( g `  y
)  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2522, 24bitrd 256 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2613, 25imbi12d 321 . . . . . 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 3184 . . . . 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 17 . . . 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 32 . . 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 1199 . 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 5891 . . . 4  |-  ( g `
 { z  |  ( F `  K
) x z } )  e.  _V
32 nfcv 2591 . . . . 5  |-  F/_ y
s
33 nfcv 2591 . . . . 5  |-  F/_ y K
34 nfcv 2591 . . . . 5  |-  F/_ y
( g `  {
z  |  ( F `
 K ) x z } )
35 axdclem.1 . . . . 5  |-  F  =  ( rec ( ( y  e.  _V  |->  ( g `  { z  |  y x z } ) ) ,  s )  |`  om )
36 breq1 4429 . . . . . . 7  |-  ( y  =  ( F `  K )  ->  (
y x z  <->  ( F `  K ) x z ) )
3736abbidv 2565 . . . . . 6  |-  ( y  =  ( F `  K )  ->  { z  |  y x z }  =  { z  |  ( F `  K ) x z } )
3837fveq2d 5885 . . . . 5  |-  ( y  =  ( F `  K )  ->  (
g `  { z  |  y x z } )  =  ( g `  { z  |  ( F `  K ) x z } ) )
3932, 33, 34, 35, 38frsucmpt 7163 . . . 4  |-  ( ( K  e.  om  /\  ( g `  {
z  |  ( F `
 K ) x z } )  e. 
_V )  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
4031, 39mpan2 675 . . 3  |-  ( K  e.  om  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
4140breq2d 4438 . 2  |-  ( K  e.  om  ->  (
( F `  K
) x ( F `
 suc  K )  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
4230, 41syl5ibrcom 225 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414    =/= wne 2625   A.wral 2782   _Vcvv 3087    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854   ran crn 4855    |` cres 4856   suc csuc 5444   ` cfv 5601   omcom 6706   reccrdg 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136
This theorem is referenced by:  axdclem2  8948
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