Theorem txopn 13768

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 2177	. . . . . 6 |- ran ( u e. R , v e. S |-> ( u X. v ) ) = ran ( u e. R , v e. S |-> ( u X. v ) )
2	1	txbasex 13760	. . . . 5 |- ( ( R e. V /\ S e. W ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V )
3		bastg 13564	. . . . 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 2177	. . . . . 6 |- ( A X. B ) = ( A X. B )
7		xpeq1 4641	. . . . . . . 8 |- ( u = A -> ( u X. v ) = ( A X. v ) )
8	7	eqeq2d 2189	. . . . . . 7 |- ( u = A -> ( ( A X. B ) = ( u X. v ) <-> ( A X. B ) = ( A X. v ) ) )
9		xpeq2 4642	. . . . . . . 8 |- ( v = B -> ( A X. v ) = ( A X. B ) )
10	9	eqeq2d 2189	. . . . . . 7 |- ( v = B -> ( ( A X. B ) = ( A X. v ) <-> ( A X. B ) = ( A X. B ) ) )
11	8, 10	rspc2ev 2857	. . . . . 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 1326	. . . . 5 |- ( ( A e. R /\ B e. S ) -> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) )
13		xpexg 4741	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> ( A X. B ) e. _V )
14		eqid 2177	. . . . . . 7 |- ( u e. R , v e. S |-> ( u X. v ) ) = ( u e. R , v e. S |-> ( u X. v ) )
15	14	elrnmpog 5987	. . . . . 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 3157	. 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 13758	. . 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 2257	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 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2738   C_ wss 3130   X. cxp 4625   ran crn 4628   ` cfv 5217  (class class class)co 5875   e. cmpo 5877   topGenctg 12703   tX ctx 13755

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-topgen 12709  df-tx 13756

This theorem is referenced by:  txbasval  13770  neitx  13771  tx1cn  13772  tx2cn  13773  txlm  13782