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Theorem axdc4uz 11277
Description: A version of axdc4 8292 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 2465 . . . . 5  |-  ( f  =  A  ->  ( C  e.  f  <->  C  e.  A ) )
2 xpeq2 4852 . . . . . 6  |-  ( f  =  A  ->  ( Z  X.  f )  =  ( Z  X.  A
) )
3 pweq 3762 . . . . . . 7  |-  ( f  =  A  ->  ~P f  =  ~P A
)
43difeq1d 3424 . . . . . 6  |-  ( f  =  A  ->  ( ~P f  \  { (/) } )  =  ( ~P A  \  { (/) } ) )
52, 4feq23d 5547 . . . . 5  |-  ( f  =  A  ->  ( F : ( Z  X.  f ) --> ( ~P f  \  { (/) } )  <->  F : ( Z  X.  A ) --> ( ~P A  \  { (/)
} ) ) )
61, 5anbi12d 692 . . . 4  |-  ( f  =  A  ->  (
( C  e.  f  /\  F : ( Z  X.  f ) --> ( ~P f  \  { (/) } ) )  <-> 
( C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/)
} ) ) ) )
7 feq3 5537 . . . . . 6  |-  ( f  =  A  ->  (
g : Z --> f  <->  g : Z
--> A ) )
873anbi1d 1258 . . . . 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 1633 . . . 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 312 . . 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 2919 . . . 4  |-  f  e. 
_V
14 eqid 2404 . . . 4  |-  ( rec ( ( y  e. 
_V  |->  ( y  +  1 ) ) ,  M )  |`  om )  =  ( rec (
( y  e.  _V  |->  ( y  +  1 ) ) ,  M
)  |`  om )
15 eqid 2404 . . . 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 11276 . . 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 2971 . 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 1151 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277   (/)c0 3588   ~Pcpw 3759   {csn 3774    e. cmpt 4226   omcom 4804    X. cxp 4835    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   reccrdg 6626   1c1 8947    + caddc 8949   ZZcz 10238   ZZ>=cuz 10444
This theorem is referenced by:  bcthlem5  19234  sdclem1  26337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-dc 8282  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445
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