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Theorem txval 17275
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 2809 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 elex 2809 . 2  |-  ( S  e.  W  ->  S  e.  _V )
3 mpt2eq12 5924 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) )  =  ( x  e.  R , 
y  e.  S  |->  ( x  X.  y ) ) )
43rneqd 4922 . . . . 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 2346 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  B )
76fveq2d 5545 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  =  ( topGen `  B ) )
8 df-tx 17273 . . 3  |-  tX  =  ( r  e.  _V ,  s  e.  _V  |->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
9 fvex 5555 . . 3  |-  ( topGen `  B )  e.  _V
107, 8, 9ovmpt2a 5994 . 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 1632    e. wcel 1696   _Vcvv 2801    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   topGenctg 13358    tX ctx 17271
This theorem is referenced by:  eltx  17279  txtop  17280  txtopon  17302  txopn  17313  txss12  17316  txbasval  17317  txcnp  17330  txcnmpt  17334  txrest  17341  txlm  17358  tx2ndc  17361  txflf  17717  mbfimaopnlem  19026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tx 17273
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