Theorem txopn 15292

Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
txopn	|- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( A X. B ) e. ( R tX S ) )

Proof of Theorem txopn

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . . 6 |- ran ( u e. R , v e. S |-> ( u X. v ) ) = ran ( u e. R , v e. S |-> ( u X. v ) )
2	1	txbasex 15284	. . . . 5 |- ( ( R e. V /\ S e. W ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V )
3		bastg 15088	. . . . 5 |- ( ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V -> ran ( u e. R , v e. S |-> ( u X. v ) ) C_ ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
4	2, 3	syl 14	. . . 4 |- ( ( R e. V /\ S e. W ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) C_ ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
5	4	adantr 276	. . 3 |- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) C_ ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
6		eqid 2238	. . . . . 6 |- ( A X. B ) = ( A X. B )
7		xpeq1 4786	. . . . . . . 8 |- ( u = A -> ( u X. v ) = ( A X. v ) )
8	7	eqeq2d 2250	. . . . . . 7 |- ( u = A -> ( ( A X. B ) = ( u X. v ) <-> ( A X. B ) = ( A X. v ) ) )
9		xpeq2 4787	. . . . . . . 8 |- ( v = B -> ( A X. v ) = ( A X. B ) )
10	9	eqeq2d 2250	. . . . . . 7 |- ( v = B -> ( ( A X. B ) = ( A X. v ) <-> ( A X. B ) = ( A X. B ) ) )
11	8, 10	rspc2ev 2945	. . . . . 6 |- ( ( A e. R /\ B e. S /\ ( A X. B ) = ( A X. B ) ) -> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) )
12	6, 11	mp3an3 1367	. . . . 5 |- ( ( A e. R /\ B e. S ) -> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) )
13		xpexg 4887	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> ( A X. B ) e. _V )
14		eqid 2238	. . . . . . 7 |- ( u e. R , v e. S |-> ( u X. v ) ) = ( u e. R , v e. S |-> ( u X. v ) )
15	14	elrnmpog 6194	. . . . . 6 |- ( ( A X. B ) e. _V -> ( ( A X. B ) e. ran ( u e. R , v e. S |-> ( u X. v ) ) <-> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) ) )
16	13, 15	syl 14	. . . . 5 |- ( ( A e. R /\ B e. S ) -> ( ( A X. B ) e. ran ( u e. R , v e. S |-> ( u X. v ) ) <-> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) ) )
17	12, 16	mpbird 167	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( A X. B ) e. ran ( u e. R , v e. S |-> ( u X. v ) ) )
18	17	adantl 277	. . 3 |- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( A X. B ) e. ran ( u e. R , v e. S |-> ( u X. v ) ) )
19	5, 18	sseldd 3249	. 2 |- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( A X. B ) e. ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
20	1	txval 15282	. . 3 |- ( ( R e. V /\ S e. W ) -> ( R tX S ) = ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
21	20	adantr 276	. 2 |- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( R tX S ) = ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
22	19, 21	eleqtrrd 2318	1 |- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( A X. B ) e. ( R tX S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   _Vcvv 2821   C_ wss 3220   X. cxp 4770   ran crn 4773   ` cfv 5375  (class class class)co 6078   e. cmpo 6080   topGenctg 13588   tX ctx 15279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-topgen 13594  df-tx 15280

This theorem is referenced by:  txbasval  15294  neitx  15295  tx1cn  15296  tx2cn  15297  txlm  15306