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Theorem axdc4uz 11322
Description: A version of axdc4 8336 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 2497 . . . . 5  |-  ( f  =  A  ->  ( C  e.  f  <->  C  e.  A ) )
2 xpeq2 4893 . . . . . 6  |-  ( f  =  A  ->  ( Z  X.  f )  =  ( Z  X.  A
) )
3 pweq 3802 . . . . . . 7  |-  ( f  =  A  ->  ~P f  =  ~P A
)
43difeq1d 3464 . . . . . 6  |-  ( f  =  A  ->  ( ~P f  \  { (/) } )  =  ( ~P A  \  { (/) } ) )
52, 4feq23d 5588 . . . . 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 5578 . . . . . 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 1636 . . . 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 2959 . . . 4  |-  f  e. 
_V
14 eqid 2436 . . . 4  |-  ( rec ( ( y  e. 
_V  |->  ( y  +  1 ) ) ,  M )  |`  om )  =  ( rec (
( y  e.  _V  |->  ( y  +  1 ) ) ,  M
)  |`  om )
15 eqid 2436 . . . 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 11321 . . 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 3011 . 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 1550    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    \ cdif 3317   (/)c0 3628   ~Pcpw 3799   {csn 3814    e. cmpt 4266   omcom 4845    X. cxp 4876    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   reccrdg 6667   1c1 8991    + caddc 8993   ZZcz 10282   ZZ>=cuz 10488
This theorem is referenced by:  bcthlem5  19281  sdclem1  26447
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-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-dc 8326  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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-nel 2602  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-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489
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