Theorem txtopon 12673

Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
txtopon	|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )

Proof of Theorem txtopon

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

Step	Hyp	Ref	Expression
1		topontop 12423	. . 3 |- ( R e. (TopOn ` X ) -> R e. Top )
2		topontop 12423	. . 3 |- ( S e. (TopOn ` Y ) -> S e. Top )
3		txtop 12671	. . 3 |- ( ( R e. Top /\ S e. Top ) -> ( R tX S ) e. Top )
4	1, 2, 3	syl2an 287	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. Top )
5		eqid 2157	. . . . 5 |- ran ( u e. R , v e. S |-> ( u X. v ) ) = ran ( u e. R , v e. S |-> ( u X. v ) )
6		eqid 2157	. . . . 5 |- U. R = U. R
7		eqid 2157	. . . . 5 |- U. S = U. S
8	5, 6, 7	txuni2 12667	. . . 4 |- ( U. R X. U. S ) = U. ran ( u e. R , v e. S |-> ( u X. v ) )
9		toponuni 12424	. . . . 5 |- ( R e. (TopOn ` X ) -> X = U. R )
10		toponuni 12424	. . . . 5 |- ( S e. (TopOn ` Y ) -> Y = U. S )
11		xpeq12 4605	. . . . 5 |- ( ( X = U. R /\ Y = U. S ) -> ( X X. Y ) = ( U. R X. U. S ) )
12	9, 10, 11	syl2an 287	. . . 4 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( X X. Y ) = ( U. R X. U. S ) )
13	5	txbasex 12668	. . . . 5 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V )
14		unitg 12473	. . . . 5 |- ( ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V -> U. ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) = U. ran ( u e. R , v e. S |-> ( u X. v ) ) )
15	13, 14	syl 14	. . . 4 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> U. ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) = U. ran ( u e. R , v e. S |-> ( u X. v ) ) )
16	8, 12, 15	3eqtr4a 2216	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( X X. Y ) = U. ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
17	5	txval 12666	. . . 4 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) = ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
18	17	unieqd 3783	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> U. ( R tX S ) = U. ( topGen ` ran ( u e. R , v e. S |-> ( u X. v ) ) ) )
19	16, 18	eqtr4d 2193	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( X X. Y ) = U. ( R tX S ) )
20		istopon 12422	. 2 |- ( ( R tX S ) e. (TopOn ` ( X X. Y ) ) <-> ( ( R tX S ) e. Top /\ ( X X. Y ) = U. ( R tX S ) ) )
21	4, 19, 20	sylanbrc 414	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   _Vcvv 2712   U.cuni 3772   X. cxp 4584   ran crn 4587   ` cfv 5170  (class class class)co 5824   e. cmpo 5826   topGenctg 12377   Topctop 12406  TopOnctopon 12419   tX ctx 12663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-topgen 12383  df-top 12407  df-topon 12420  df-bases 12452  df-tx 12664

This theorem is referenced by:  txuni  12674  tx1cn  12680  tx2cn  12681  txcnp  12682  txcnmpt  12684  txdis1cn  12689  txlm  12690  lmcn2  12691  cnmpt12  12698  cnmpt2c  12701  cnmpt21  12702  cnmpt2t  12704  cnmpt22  12705  cnmpt22f  12706  cnmpt2res  12708  cnmptcom  12709  txmetcn  12930  limccnp2lem  13056  limccnp2cntop  13057  dvcnp2cntop  13074  dvaddxxbr  13076  dvmulxxbr  13077  dvcoapbr  13082