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Theorem txval 17259
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 2796 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 elex 2796 . 2  |-  ( S  e.  W  ->  S  e.  _V )
3 mpt2eq12 5908 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) )  =  ( x  e.  R , 
y  e.  S  |->  ( x  X.  y ) ) )
43rneqd 4906 . . . . 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 2333 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  B )
76fveq2d 5529 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  =  ( topGen `  B ) )
8 df-tx 17257 . . 3  |-  tX  =  ( r  e.  _V ,  s  e.  _V  |->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
9 fvex 5539 . . 3  |-  ( topGen `  B )  e.  _V
107, 8, 9ovmpt2a 5978 . 2  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
111, 2, 10syl2an 463 1  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   topGenctg 13342    tX ctx 17255
This theorem is referenced by:  eltx  17263  txtop  17264  txtopon  17286  txopn  17297  txss12  17300  txbasval  17301  txcnp  17314  txcnmpt  17318  txrest  17325  txlm  17342  tx2ndc  17345  txflf  17701  mbfimaopnlem  19010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tx 17257
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