Theorem txtopon 14952

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 14704	. . 3 |- ( R e. (TopOn ` X ) -> R e. Top )
2		topontop 14704	. . 3 |- ( S e. (TopOn ` Y ) -> S e. Top )
3		txtop 14950	. . 3 |- ( ( R e. Top /\ S e. Top ) -> ( R tX S ) e. Top )
4	1, 2, 3	syl2an 289	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. Top )
5		eqid 2229	. . . . 5 |- ran ( u e. R , v e. S |-> ( u X. v ) ) = ran ( u e. R , v e. S |-> ( u X. v ) )
6		eqid 2229	. . . . 5 |- U. R = U. R
7		eqid 2229	. . . . 5 |- U. S = U. S
8	5, 6, 7	txuni2 14946	. . . 4 |- ( U. R X. U. S ) = U. ran ( u e. R , v e. S |-> ( u X. v ) )
9		toponuni 14705	. . . . 5 |- ( R e. (TopOn ` X ) -> X = U. R )
10		toponuni 14705	. . . . 5 |- ( S e. (TopOn ` Y ) -> Y = U. S )
11		xpeq12 4738	. . . . 5 |- ( ( X = U. R /\ Y = U. S ) -> ( X X. Y ) = ( U. R X. U. S ) )
12	9, 10, 11	syl2an 289	. . . 4 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( X X. Y ) = ( U. R X. U. S ) )
13	5	txbasex 14947	. . . . 5 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ran ( u e. R , v e. S |-> ( u X. v ) ) e. _V )
14		unitg 14752	. . . . 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 2288	. . 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 14945	. . . 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 3899	. . 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 2265	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( X X. Y ) = U. ( R tX S ) )
20		istopon 14703	. 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 417	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   U.cuni 3888   X. cxp 4717   ran crn 4720   ` cfv 5318  (class class class)co 6007   e. cmpo 6009   topGenctg 13303   Topctop 14687  TopOnctopon 14700   tX ctx 14942

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 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-topgen 13309  df-top 14688  df-topon 14701  df-bases 14733  df-tx 14943

This theorem is referenced by:  txuni  14953  tx1cn  14959  tx2cn  14960  txcnp  14961  txcnmpt  14963  txdis1cn  14968  txlm  14969  lmcn2  14970  cnmpt12  14977  cnmpt2c  14980  cnmpt21  14981  cnmpt2t  14983  cnmpt22  14984  cnmpt22f  14985  cnmpt2res  14987  cnmptcom  14988  txmetcn  15209  limccnp2lem  15366  limccnp2cntop  15367  dvcnp2cntop  15389  dvaddxxbr  15391  dvmulxxbr  15392  dvcoapbr  15397