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Theorem prodex 25235
Description: A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
Assertion
Ref Expression
prodex  |-  prod_ k  e.  A B  e.  _V

Proof of Theorem prodex
Dummy variables  f  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 25234 . 2  |-  prod_ k  e.  A B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  E. n  e.  ( ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq  m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
2 iotaex 5437 . 2  |-  ( iota
x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  (
ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq  m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )  e.  _V
31, 2eqeltri 2508 1  |-  prod_ k  e.  A B  e.  _V
Colors of variables: wff set class
Syntax hints:    \/ wo 359    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   _Vcvv 2958   [_csb 3253    C_ wss 3322   ifcif 3741   class class class wbr 4214    e. cmpt 4268   iotacio 5418   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    x. cmul 8997   NNcn 10002   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045    seq cseq 11325    ~~> cli 12280   prod_cprod 25233
This theorem is referenced by:  risefacval  25326  fallfacval  25327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5420  df-prod 25234
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