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Theorem axdc4uz 11045
Description: A version of axdc4 8082 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 2344 . . . . 5  |-  ( f  =  A  ->  ( C  e.  f  <->  C  e.  A ) )
2 xpeq2 4704 . . . . . 6  |-  ( f  =  A  ->  ( Z  X.  f )  =  ( Z  X.  A
) )
3 pweq 3628 . . . . . . 7  |-  ( f  =  A  ->  ~P f  =  ~P A
)
43difeq1d 3293 . . . . . 6  |-  ( f  =  A  ->  ( ~P f  \  { (/) } )  =  ( ~P A  \  { (/) } ) )
52, 4feq23d 5386 . . . . 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 5377 . . . . . 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 1612 . . . 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 2791 . . . 4  |-  f  e. 
_V
14 eqid 2283 . . . 4  |-  ( rec ( ( y  e. 
_V  |->  ( y  +  1 ) ) ,  M )  |`  om )  =  ( rec (
( y  e.  _V  |->  ( y  +  1 ) ) ,  M
)  |`  om )
15 eqid 2283 . . . 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 11044 . . 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 2843 . 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 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640    e. cmpt 4077   omcom 4656    X. cxp 4687    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   reccrdg 6422   1c1 8738    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230
This theorem is referenced by:  bcthlem5  18750  sdclem1  25865
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-dc 8072  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-nel 2449  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231
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