Theorem txopn 14852

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 2207	. . . . . 6 |- ran ( u e. R , v e. S |-> ( u X. v ) ) = ran ( u e. R , v e. S |-> ( u X. v ) )
2	1	txbasex 14844	. . . . 5 |- ( ( R e. V /\ S e. W ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V )
3		bastg 14648	. . . . 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 2207	. . . . . 6 |- ( A X. B ) = ( A X. B )
7		xpeq1 4707	. . . . . . . 8 |- ( u = A -> ( u X. v ) = ( A X. v ) )
8	7	eqeq2d 2219	. . . . . . 7 |- ( u = A -> ( ( A X. B ) = ( u X. v ) <-> ( A X. B ) = ( A X. v ) ) )
9		xpeq2 4708	. . . . . . . 8 |- ( v = B -> ( A X. v ) = ( A X. B ) )
10	9	eqeq2d 2219	. . . . . . 7 |- ( v = B -> ( ( A X. B ) = ( A X. v ) <-> ( A X. B ) = ( A X. B ) ) )
11	8, 10	rspc2ev 2899	. . . . . 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 1339	. . . . 5 |- ( ( A e. R /\ B e. S ) -> E. u e. R E. v e. S ( A X. B ) = ( u X. v ) )
13		xpexg 4807	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> ( A X. B ) e. _V )
14		eqid 2207	. . . . . . 7 |- ( u e. R , v e. S |-> ( u X. v ) ) = ( u e. R , v e. S |-> ( u X. v ) )
15	14	elrnmpog 6081	. . . . . 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 3202	. 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 14842	. . 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 2287	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 1373   e. wcel 2178   E.wrex 2487   _Vcvv 2776   C_ wss 3174   X. cxp 4691   ran crn 4694   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   topGenctg 13201   tX ctx 14839

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-topgen 13207  df-tx 14840

This theorem is referenced by:  txbasval  14854  neitx  14855  tx1cn  14856  tx2cn  14857  txlm  14866