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Theorem axdclem 8399
Description: Lemma for axdc 8401. (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 5742 . . . . . . . . 9  |-  ( F `
 K )  e. 
_V
2 vex 2959 . . . . . . . . 9  |-  z  e. 
_V
31, 2brelrn 5100 . . . . . . . 8  |-  ( ( F `  K ) x z  ->  z  e.  ran  x )
43abssi 3418 . . . . . . 7  |-  { z  |  ( F `  K ) x z }  C_  ran  x
5 sstr 3356 . . . . . . 7  |-  ( ( { z  |  ( F `  K ) x z }  C_  ran  x  /\  ran  x  C_ 
dom  x )  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
64, 5mpan 652 . . . . . 6  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
7 vex 2959 . . . . . . . 8  |-  x  e. 
_V
87dmex 5132 . . . . . . 7  |-  dom  x  e.  _V
98elpw2 4364 . . . . . 6  |-  ( { z  |  ( F `
 K ) x z }  e.  ~P dom  x  <->  { z  |  ( F `  K ) x z }  C_  dom  x )
106, 9sylibr 204 . . . . 5  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  e.  ~P dom  x )
11 neeq1 2609 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  { z  |  ( F `  K ) x z }  =/=  (/) ) )
12 abn0 3646 . . . . . . . 8  |-  ( { z  |  ( F `
 K ) x z }  =/=  (/)  <->  E. z
( F `  K
) x z )
1311, 12syl6bb 253 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  E. z ( F `
 K ) x z ) )
14 eleq2 2497 . . . . . . . . . 10  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
z  |  ( F `
 K ) x z } ) )
15 breq2 4216 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
( F `  K
) x w  <->  ( F `  K ) x z ) )
1615cbvabv 2555 . . . . . . . . . . 11  |-  { w  |  ( F `  K ) x w }  =  { z  |  ( F `  K ) x z }
1716eleq2i 2500 . . . . . . . . . 10  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( g `  y )  e.  { z  |  ( F `  K
) x z } )
1814, 17syl6bbr 255 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
w  |  ( F `
 K ) x w } ) )
19 fvex 5742 . . . . . . . . . 10  |-  ( g `
 y )  e. 
_V
20 breq2 4216 . . . . . . . . . 10  |-  ( w  =  ( g `  y )  ->  (
( F `  K
) x w  <->  ( F `  K ) x ( g `  y ) ) )
2119, 20elab 3082 . . . . . . . . 9  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( F `  K ) x ( g `  y ) )
2218, 21syl6bb 253 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  y ) ) )
23 fveq2 5728 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( g `  y )  =  ( g `  { z  |  ( F `  K ) x z } ) )
2423breq2d 4224 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( F `
 K ) x ( g `  y
)  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2522, 24bitrd 245 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2613, 25imbi12d 312 . . . . . 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 3048 . . . . 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 16 . . . 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 29 . . 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 1147 . 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 5742 . . . 4  |-  ( g `
 { z  |  ( F `  K
) x z } )  e.  _V
32 nfcv 2572 . . . . 5  |-  F/_ y
s
33 nfcv 2572 . . . . 5  |-  F/_ y K
34 nfcv 2572 . . . . 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 4215 . . . . . . 7  |-  ( y  =  ( F `  K )  ->  (
y x z  <->  ( F `  K ) x z ) )
3736abbidv 2550 . . . . . 6  |-  ( y  =  ( F `  K )  ->  { z  |  y x z }  =  { z  |  ( F `  K ) x z } )
3837fveq2d 5732 . . . . 5  |-  ( y  =  ( F `  K )  ->  (
g `  { z  |  y x z } )  =  ( g `  { z  |  ( F `  K ) x z } ) )
3932, 33, 34, 35, 38frsucmpt 6695 . . . 4  |-  ( ( K  e.  om  /\  ( g `  {
z  |  ( F `
 K ) x z } )  e. 
_V )  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
4031, 39mpan2 653 . . 3  |-  ( K  e.  om  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
4140breq2d 4224 . 2  |-  ( K  e.  om  ->  (
( F `  K
) x ( F `
 suc  K )  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
4230, 41syl5ibrcom 214 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 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   class class class wbr 4212    e. cmpt 4266   suc csuc 4583   omcom 4845   dom cdm 4878   ran crn 4879    |` cres 4880   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  axdclem2  8400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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