Theorem txopn 14939

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 2229	. . . . . 6 |- ran ( u e. R , v e. S |-> ( u X. v ) ) = ran ( u e. R , v e. S |-> ( u X. v ) )
2	1	txbasex 14931	. . . . 5 |- ( ( R e. V /\ S e. W ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V )
3		bastg 14735	. . . . 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 2229	. . . . . 6 |- ( A X. B ) = ( A X. B )
7		xpeq1 4733	. . . . . . . 8 |- ( u = A -> ( u X. v ) = ( A X. v ) )
8	7	eqeq2d 2241	. . . . . . 7 |- ( u = A -> ( ( A X. B ) = ( u X. v ) <-> ( A X. B ) = ( A X. v ) ) )
9		xpeq2 4734	. . . . . . . 8 |- ( v = B -> ( A X. v ) = ( A X. B ) )
10	9	eqeq2d 2241	. . . . . . 7 |- ( v = B -> ( ( A X. B ) = ( A X. v ) <-> ( A X. B ) = ( A X. B ) ) )
11	8, 10	rspc2ev 2922	. . . . . 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 1360	. . . . 5 |- ( ( A e. R /\ B e. S ) -> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) )
13		xpexg 4833	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> ( A X. B ) e. _V )
14		eqid 2229	. . . . . . 7 |- ( u e. R , v e. S |-> ( u X. v ) ) = ( u e. R , v e. S |-> ( u X. v ) )
15	14	elrnmpog 6117	. . . . . 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 3225	. 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 14929	. . 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 2309	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 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   C_ wss 3197   X. cxp 4717   ran crn 4720   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   topGenctg 13287   tX ctx 14926

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-topgen 13293  df-tx 14927

This theorem is referenced by:  txbasval  14941  neitx  14942  tx1cn  14943  tx2cn  14944  txlm  14953