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Theorem rabrenfdioph 26566
Description: Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Assertion
Ref Expression
rabrenfdioph  |-  ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B )  /\  { a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph }  e.  (Dioph `  A ) )  ->  { b  e.  ( NN0  ^m  ( 1 ... B ) )  |  [. ( b  o.  F )  / 
a ]. ph }  e.  (Dioph `  B ) )
Distinct variable groups:    ph, b    A, a, b    B, a, b    F, a, b
Allowed substitution hint:    ph( a)

Proof of Theorem rabrenfdioph
StepHypRef Expression
1 simpr 448 . . . . . . 7  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  b  e.  ( NN0  ^m  ( 1 ... B ) ) )
2 simplr 732 . . . . . . 7  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  F :
( 1 ... A
) --> ( 1 ... B ) )
3 ovex 6045 . . . . . . . 8  |-  ( 1 ... A )  e. 
_V
43mapco2 26461 . . . . . . 7  |-  ( ( b  e.  ( NN0 
^m  ( 1 ... B ) )  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  ->  ( b  o.  F )  e.  ( NN0  ^m  ( 1 ... A ) ) )
51, 2, 4syl2anc 643 . . . . . 6  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  ( b  o.  F )  e.  ( NN0  ^m  ( 1 ... A ) ) )
65biantrurd 495 . . . . 5  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  ( [. ( b  o.  F
)  /  a ]. ph  <->  ( ( b  o.  F
)  e.  ( NN0 
^m  ( 1 ... A ) )  /\  [. ( b  o.  F
)  /  a ]. ph ) ) )
7 nfcv 2523 . . . . . 6  |-  F/_ a
( NN0  ^m  (
1 ... A ) )
87elrabsf 3142 . . . . 5  |-  ( ( b  o.  F )  e.  { a  e.  ( NN0  ^m  (
1 ... A ) )  |  ph }  <->  ( (
b  o.  F )  e.  ( NN0  ^m  ( 1 ... A
) )  /\  [. (
b  o.  F )  /  a ]. ph )
)
96, 8syl6bbr 255 . . . 4  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  ( [. ( b  o.  F
)  /  a ]. ph  <->  ( b  o.  F )  e.  { a  e.  ( NN0  ^m  (
1 ... A ) )  |  ph } ) )
109rabbidva 2890 . . 3  |-  ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  ->  { b  e.  ( NN0  ^m  (
1 ... B ) )  |  [. ( b  o.  F )  / 
a ]. ph }  =  { b  e.  ( NN0  ^m  ( 1 ... B ) )  |  ( b  o.  F )  e.  {
a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph } } )
11103adant3 977 . 2  |-  ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B )  /\  { a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph }  e.  (Dioph `  A ) )  ->  { b  e.  ( NN0  ^m  ( 1 ... B ) )  |  [. ( b  o.  F )  / 
a ]. ph }  =  { b  e.  ( NN0  ^m  ( 1 ... B ) )  |  ( b  o.  F )  e.  {
a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph } } )
12 diophren 26565 . . 3  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... A ) )  |  ph }  e.  (Dioph `  A )  /\  B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  ->  { b  e.  ( NN0  ^m  (
1 ... B ) )  |  ( b  o.  F )  e.  {
a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph } }  e.  (Dioph `  B ) )
13123coml 1160 . 2  |-  ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B )  /\  { a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph }  e.  (Dioph `  A ) )  ->  { b  e.  ( NN0  ^m  ( 1 ... B ) )  |  ( b  o.  F )  e.  {
a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph } }  e.  (Dioph `  B ) )
1411, 13eqeltrd 2461 1  |-  ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B )  /\  { a  e.  ( NN0 
^m  ( 1 ... A ) )  | 
ph }  e.  (Dioph `  A ) )  ->  { b  e.  ( NN0  ^m  ( 1 ... B ) )  |  [. ( b  o.  F )  / 
a ]. ph }  e.  (Dioph `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2653   [.wsbc 3104    o. ccom 4822   -->wf 5390   ` cfv 5394  (class class class)co 6020    ^m cmap 6954   1c1 8924   NN0cn0 10153   ...cfz 10975  Diophcdioph 26504
This theorem is referenced by:  rabren3dioph  26567
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-hash 11546  df-mzpcl 26471  df-mzp 26472  df-dioph 26505
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