Theorem prodex 14637

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.

Step	Hyp	Ref	Expression
1		df-prod 14636	. 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 5868	. 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
3	1, 2	eqeltri 2697	1 |- prod_ k e. A B e. _V

Syntax hints:   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   [_csb 3533   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   iotacio 5849   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-prod 14636

This theorem is referenced by:  risefacval  14739  fallfacval  14740  prmoval  15737  fprodsubrecnncnvlem  40121  fprodaddrecnncnvlem  40123  etransclem13  40464  ovnlecvr  40772  ovncvrrp  40778  hoidmvval  40791  vonioolem1  40894