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Theorem rabrenfdioph 30579
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 461 . . . . . . 7  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  b  e.  ( NN0  ^m  ( 1 ... B ) ) )
2 simplr 754 . . . . . . 7  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  F :
( 1 ... A
) --> ( 1 ... B ) )
3 ovex 6310 . . . . . . . 8  |-  ( 1 ... A )  e. 
_V
43mapco2 30478 . . . . . . 7  |-  ( ( b  e.  ( NN0 
^m  ( 1 ... B ) )  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  ->  ( b  o.  F )  e.  ( NN0  ^m  ( 1 ... A ) ) )
51, 2, 4syl2anc 661 . . . . . 6  |-  ( ( ( B  e.  NN0  /\  F : ( 1 ... A ) --> ( 1 ... B ) )  /\  b  e.  ( NN0  ^m  (
1 ... B ) ) )  ->  ( b  o.  F )  e.  ( NN0  ^m  ( 1 ... A ) ) )
65biantrurd 508 . . . . 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 2629 . . . . . 6  |-  F/_ a
( NN0  ^m  (
1 ... A ) )
87elrabsf 3370 . . . . 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 263 . . . 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 3104 . . 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 1016 . 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 30578 . . 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 1203 . 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 2555 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   [.wsbc 3331    o. ccom 5003   -->wf 5584   ` cfv 5588  (class class class)co 6285    ^m cmap 7421   1c1 9494   NN0cn0 10796   ...cfz 11673  Diophcdioph 30519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-hash 12375  df-mzpcl 30486  df-mzp 30487  df-dioph 30520
This theorem is referenced by:  rabren3dioph  30580
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