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Theorem txval 19140
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 2984 . 2  |-  ( R  e.  V  ->  R  e.  _V )
2 elex 2984 . 2  |-  ( S  e.  W  ->  S  e.  _V )
3 mpt2eq12 6149 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) )  =  ( x  e.  R , 
y  e.  S  |->  ( x  X.  y ) ) )
43rneqd 5070 . . . . 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 2493 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  B )
76fveq2d 5698 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  =  ( topGen `  B ) )
8 df-tx 19138 . . 3  |-  tX  =  ( r  e.  _V ,  s  e.  _V  |->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
9 fvex 5704 . . 3  |-  ( topGen `  B )  e.  _V
107, 8, 9ovmpt2a 6224 . 2  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  tX  S
)  =  ( topGen `  B ) )
111, 2, 10syl2an 477 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 1369    e. wcel 1756   _Vcvv 2975    X. cxp 4841   ran crn 4844   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   topGenctg 14379    tX ctx 19136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pr 4534
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-iota 5384  df-fun 5423  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-tx 19138
This theorem is referenced by:  eltx  19144  txtop  19145  txtopon  19167  txopn  19178  txss12  19181  txbasval  19182  txcnp  19196  txcnmpt  19200  txrest  19207  txlm  19224  tx2ndc  19227  txflf  19582  mbfimaopnlem  21136
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