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Theorem axdc4uz 11061
Description: A version of axdc4 8098 that works on a set of upper integers instead of  om. (Contributed by Mario Carneiro, 8-Jan-2014.)
Hypotheses
Ref Expression
axdc4uz.1  |-  M  e.  ZZ
axdc4uz.2  |-  Z  =  ( ZZ>= `  M )
Assertion
Ref Expression
axdc4uz  |-  ( ( A  e.  V  /\  C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/) } ) )  ->  E. g
( g : Z --> A  /\  ( g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) ) )
Distinct variable groups:    g, k, A    C, g    g, F, k    g, M, k   
g, Z
Allowed substitution hints:    C( k)    V( g, k)    Z( k)

Proof of Theorem axdc4uz
Dummy variables  f  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2357 . . . . 5  |-  ( f  =  A  ->  ( C  e.  f  <->  C  e.  A ) )
2 xpeq2 4720 . . . . . 6  |-  ( f  =  A  ->  ( Z  X.  f )  =  ( Z  X.  A
) )
3 pweq 3641 . . . . . . 7  |-  ( f  =  A  ->  ~P f  =  ~P A
)
43difeq1d 3306 . . . . . 6  |-  ( f  =  A  ->  ( ~P f  \  { (/) } )  =  ( ~P A  \  { (/) } ) )
52, 4feq23d 5402 . . . . 5  |-  ( f  =  A  ->  ( F : ( Z  X.  f ) --> ( ~P f  \  { (/) } )  <->  F : ( Z  X.  A ) --> ( ~P A  \  { (/)
} ) ) )
61, 5anbi12d 691 . . . 4  |-  ( f  =  A  ->  (
( C  e.  f  /\  F : ( Z  X.  f ) --> ( ~P f  \  { (/) } ) )  <-> 
( C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/)
} ) ) ) )
7 feq3 5393 . . . . . 6  |-  ( f  =  A  ->  (
g : Z --> f  <->  g : Z
--> A ) )
873anbi1d 1256 . . . . 5  |-  ( f  =  A  ->  (
( g : Z --> f  /\  ( g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) )  <->  ( g : Z --> A  /\  (
g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) ) ) )
98exbidv 1616 . . . 4  |-  ( f  =  A  ->  ( E. g ( g : Z --> f  /\  (
g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) )  <->  E. g ( g : Z --> A  /\  ( g `  M
)  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `  k
) ) ) ) )
106, 9imbi12d 311 . . 3  |-  ( f  =  A  ->  (
( ( C  e.  f  /\  F :
( Z  X.  f
) --> ( ~P f  \  { (/) } ) )  ->  E. g ( g : Z --> f  /\  ( g `  M
)  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `  k
) ) ) )  <-> 
( ( C  e.  A  /\  F :
( Z  X.  A
) --> ( ~P A  \  { (/) } ) )  ->  E. g ( g : Z --> A  /\  ( g `  M
)  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `  k
) ) ) ) ) )
11 axdc4uz.1 . . . 4  |-  M  e.  ZZ
12 axdc4uz.2 . . . 4  |-  Z  =  ( ZZ>= `  M )
13 vex 2804 . . . 4  |-  f  e. 
_V
14 eqid 2296 . . . 4  |-  ( rec ( ( y  e. 
_V  |->  ( y  +  1 ) ) ,  M )  |`  om )  =  ( rec (
( y  e.  _V  |->  ( y  +  1 ) ) ,  M
)  |`  om )
15 eqid 2296 . . . 4  |-  ( n  e.  om ,  x  e.  f  |->  ( ( ( rec ( ( y  e.  _V  |->  ( y  +  1 ) ) ,  M )  |`  om ) `  n
) F x ) )  =  ( n  e.  om ,  x  e.  f  |->  ( ( ( rec ( ( y  e.  _V  |->  ( y  +  1 ) ) ,  M )  |`  om ) `  n
) F x ) )
1611, 12, 13, 14, 15axdc4uzlem 11060 . . 3  |-  ( ( C  e.  f  /\  F : ( Z  X.  f ) --> ( ~P f  \  { (/) } ) )  ->  E. g
( g : Z --> f  /\  ( g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) ) )
1710, 16vtoclg 2856 . 2  |-  ( A  e.  V  ->  (
( C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/)
} ) )  ->  E. g ( g : Z --> A  /\  (
g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) ) ) )
18173impib 1149 1  |-  ( ( A  e.  V  /\  C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/) } ) )  ->  E. g
( g : Z --> A  /\  ( g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `
 k ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   (/)c0 3468   ~Pcpw 3638   {csn 3653    e. cmpt 4093   omcom 4672    X. cxp 4703    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   reccrdg 6438   1c1 8754    + caddc 8756   ZZcz 10040   ZZ>=cuz 10246
This theorem is referenced by:  bcthlem5  18766  sdclem1  26556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-dc 8088  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247
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