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Theorem axdclem 8700
Description: Lemma for axdc 8702. (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 5713 . . . . . . . . 9  |-  ( F `
 K )  e. 
_V
2 vex 2987 . . . . . . . . 9  |-  z  e. 
_V
31, 2brelrn 5082 . . . . . . . 8  |-  ( ( F `  K ) x z  ->  z  e.  ran  x )
43abssi 3439 . . . . . . 7  |-  { z  |  ( F `  K ) x z }  C_  ran  x
5 sstr 3376 . . . . . . 7  |-  ( ( { z  |  ( F `  K ) x z }  C_  ran  x  /\  ran  x  C_ 
dom  x )  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
64, 5mpan 670 . . . . . 6  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  C_  dom  x )
7 vex 2987 . . . . . . . 8  |-  x  e. 
_V
87dmex 6523 . . . . . . 7  |-  dom  x  e.  _V
98elpw2 4468 . . . . . 6  |-  ( { z  |  ( F `
 K ) x z }  e.  ~P dom  x  <->  { z  |  ( F `  K ) x z }  C_  dom  x )
106, 9sylibr 212 . . . . 5  |-  ( ran  x  C_  dom  x  ->  { z  |  ( F `  K ) x z }  e.  ~P dom  x )
11 neeq1 2628 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  { z  |  ( F `  K ) x z }  =/=  (/) ) )
12 abn0 3668 . . . . . . . 8  |-  ( { z  |  ( F `
 K ) x z }  =/=  (/)  <->  E. z
( F `  K
) x z )
1311, 12syl6bb 261 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( y  =/=  (/) 
<->  E. z ( F `
 K ) x z ) )
14 eleq2 2504 . . . . . . . . . 10  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
z  |  ( F `
 K ) x z } ) )
15 breq2 4308 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
( F `  K
) x w  <->  ( F `  K ) x z ) )
1615cbvabv 2570 . . . . . . . . . . 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 263 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( g `  y )  e.  {
w  |  ( F `
 K ) x w } ) )
19 fvex 5713 . . . . . . . . . 10  |-  ( g `
 y )  e. 
_V
20 breq2 4308 . . . . . . . . . 10  |-  ( w  =  ( g `  y )  ->  (
( F `  K
) x w  <->  ( F `  K ) x ( g `  y ) ) )
2119, 20elab 3118 . . . . . . . . 9  |-  ( ( g `  y )  e.  { w  |  ( F `  K
) x w }  <->  ( F `  K ) x ( g `  y ) )
2218, 21syl6bb 261 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  y ) ) )
23 fveq2 5703 . . . . . . . . 9  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( g `  y )  =  ( g `  { z  |  ( F `  K ) x z } ) )
2423breq2d 4316 . . . . . . . 8  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( F `
 K ) x ( g `  y
)  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2522, 24bitrd 253 . . . . . . 7  |-  ( y  =  { z  |  ( F `  K
) x z }  ->  ( ( g `
 y )  e.  y  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
2613, 25imbi12d 320 . . . . . 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 3081 . . . . 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 31 . . 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 1181 . 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 5713 . . . 4  |-  ( g `
 { z  |  ( F `  K
) x z } )  e.  _V
32 nfcv 2589 . . . . 5  |-  F/_ y
s
33 nfcv 2589 . . . . 5  |-  F/_ y K
34 nfcv 2589 . . . . 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 4307 . . . . . . 7  |-  ( y  =  ( F `  K )  ->  (
y x z  <->  ( F `  K ) x z ) )
3736abbidv 2563 . . . . . 6  |-  ( y  =  ( F `  K )  ->  { z  |  y x z }  =  { z  |  ( F `  K ) x z } )
3837fveq2d 5707 . . . . 5  |-  ( y  =  ( F `  K )  ->  (
g `  { z  |  y x z } )  =  ( g `  { z  |  ( F `  K ) x z } ) )
3932, 33, 34, 35, 38frsucmpt 6905 . . . 4  |-  ( ( K  e.  om  /\  ( g `  {
z  |  ( F `
 K ) x z } )  e. 
_V )  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
4031, 39mpan2 671 . . 3  |-  ( K  e.  om  ->  ( F `  suc  K )  =  ( g `  { z  |  ( F `  K ) x z } ) )
4140breq2d 4316 . 2  |-  ( K  e.  om  ->  (
( F `  K
) x ( F `
 suc  K )  <->  ( F `  K ) x ( g `  { z  |  ( F `  K ) x z } ) ) )
4230, 41syl5ibrcom 222 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2618   A.wral 2727   _Vcvv 2984    C_ wss 3340   (/)c0 3649   ~Pcpw 3872   class class class wbr 4304    e. cmpt 4362   suc csuc 4733   dom cdm 4852   ran crn 4853    |` cres 4854   ` cfv 5430   omcom 6488   reccrdg 6877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-recs 6844  df-rdg 6878
This theorem is referenced by:  axdclem2  8701
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