MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txval Structured version   Unicode version

Theorem txval 17589
Description: Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
Hypothesis
Ref Expression
txval.1  |-  B  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) )
Assertion
Ref Expression
txval  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
Distinct variable groups:    x, y, R    x, S, y
Allowed substitution hints:    B( x, y)    V( x, y)    W( x, y)

Proof of Theorem txval
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2957 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 elex 2957 . 2  |-  ( S  e.  W  ->  S  e.  _V )
3 mpt2eq12 6127 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) )  =  ( x  e.  R , 
y  e.  S  |->  ( x  X.  y ) ) )
43rneqd 5090 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) ) )
5 txval.1 . . . . 5  |-  B  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) )
64, 5syl6eqr 2486 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  B )
76fveq2d 5725 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  =  ( topGen `  B ) )
8 df-tx 17587 . . 3  |-  tX  =  ( r  e.  _V ,  s  e.  _V  |->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
9 fvex 5735 . . 3  |-  ( topGen `  B )  e.  _V
107, 8, 9ovmpt2a 6197 . 2  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
111, 2, 10syl2an 464 1  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2949    X. cxp 4869   ran crn 4872   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076   topGenctg 13658    tX ctx 17585
This theorem is referenced by:  eltx  17593  txtop  17594  txtopon  17616  txopn  17627  txss12  17630  txbasval  17631  txcnp  17645  txcnmpt  17649  txrest  17656  txlm  17673  tx2ndc  17676  txflf  18031  mbfimaopnlem  19540
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-tx 17587
  Copyright terms: Public domain W3C validator