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

Theorem txval 18979
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 2971 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 elex 2971 . 2  |-  ( S  e.  W  ->  S  e.  _V )
3 mpt2eq12 6135 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) )  =  ( x  e.  R , 
y  e.  S  |->  ( x  X.  y ) ) )
43rneqd 5054 . . . . 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 2483 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  B )
76fveq2d 5683 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  =  ( topGen `  B ) )
8 df-tx 18977 . . 3  |-  tX  =  ( r  e.  _V ,  s  e.  _V  |->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
9 fvex 5689 . . 3  |-  ( topGen `  B )  e.  _V
107, 8, 9ovmpt2a 6210 . 2  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
111, 2, 10syl2an 474 1  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   _Vcvv 2962    X. cxp 4825   ran crn 4828   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   topGenctg 14359    tX ctx 18975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-iota 5369  df-fun 5408  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-tx 18977
This theorem is referenced by:  eltx  18983  txtop  18984  txtopon  19006  txopn  19017  txss12  19020  txbasval  19021  txcnp  19035  txcnmpt  19039  txrest  19046  txlm  19063  tx2ndc  19066  txflf  19421  mbfimaopnlem  20975
  Copyright terms: Public domain W3C validator