Theorem txuni 8935

Description: The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
txuni.1	|- T = (R X.t S)
txuni.2	|- X = U.R
txuni.3	|- Y = U.S

Assertion

Ref	Expression
txuni	|- ((R e. Top /\ S e. Top) -> U.T = (X X. Y))

Proof of Theorem txuni

Step	Hyp	Ref	Expression
1		txval 8932	. . . 4 |- ((R e. Top /\ S e. Top) -> (R X.t S) = (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
2		txuni.1	. . . 4 |- T = (R X.t S)
3	1, 2	syl5eq 1940	. . 3 |- ((R e. Top /\ S e. Top) -> T = (topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
4	3	unieqd 3188	. 2 |- ((R e. Top /\ S e. Top) -> U.T = U.(topGen` {z | E.u e. R E.v e. S z = (u X. v)}))
5		txbas 8933	. . 3 |- ((R e. Top /\ S e. Top) -> {z | E.u e. R E.v e. S z = (u X. v)} e. Bases)
6		unitg 8893	. . 3 |- ({z | E.u e. R E.v e. S z = (u X. v)} e. Bases -> U.(topGen` {z | E.u e. R E.v e. S z = (u X. v)}) = U.{z | E.u e. R E.v e. S z = (u X. v)})
7	5, 6	syl 12	. 2 |- ((R e. Top /\ S e. Top) -> U.(topGen` {z | E.u e. R E.v e. S z = (u X. v)}) = U.{z | E.u e. R E.v e. S z = (u X. v)})
8		xpss12 4089	. . . . . . . . . . 11 |- ((u C_ X /\ v C_ Y) -> (u X. v) C_ (X X. Y))
9		txuni.2	. . . . . . . . . . . 12 |- X = U.R
10	9	eltopss 8872	. . . . . . . . . . 11 |- ((R e. Top /\ u e. R) -> u C_ X)
11		txuni.3	. . . . . . . . . . . 12 |- Y = U.S
12	11	eltopss 8872	. . . . . . . . . . 11 |- ((S e. Top /\ v e. S) -> v C_ Y)
13	8, 10, 12	syl2an 503	. . . . . . . . . 10 |- (((R e. Top /\ u e. R) /\ (S e. Top /\ v e. S)) -> (u X. v) C_ (X X. Y))
14	13	an4s 566	. . . . . . . . 9 |- (((R e. Top /\ S e. Top) /\ (u e. R /\ v e. S)) -> (u X. v) C_ (X X. Y))
15	14	ex 402	. . . . . . . 8 |- ((R e. Top /\ S e. Top) -> ((u e. R /\ v e. S) -> (u X. v) C_ (X X. Y)))
16		sseq1 2637	. . . . . . . . 9 |- (x = (u X. v) -> (x C_ (X X. Y) <-> (u X. v) C_ (X X. Y)))
17	16	biimprcd 173	. . . . . . . 8 |- ((u X. v) C_ (X X. Y) -> (x = (u X. v) -> x C_ (X X. Y)))
18	15, 17	syl6 25	. . . . . . 7 |- ((R e. Top /\ S e. Top) -> ((u e. R /\ v e. S) -> (x = (u X. v) -> x C_ (X X. Y))))
19	18	r19.23advv 2218	. . . . . 6 |- ((R e. Top /\ S e. Top) -> (E.u e. R E.v e. S x = (u X. v) -> x C_ (X X. Y)))
20		visset 2295	. . . . . . 7 |- x e. _V
21		eqeq1 1890	. . . . . . . 8 |- (z = x -> (z = (u X. v) <-> x = (u X. v)))
22	21	2rexbidv 2141	. . . . . . 7 |- (z = x -> (E.u e. R E.v e. S z = (u X. v) <-> E.u e. R E.v e. S x = (u X. v)))
23	20, 22	elab 2403	. . . . . 6 |- (x e. {z | E.u e. R E.v e. S z = (u X. v)} <-> E.u e. R E.v e. S x = (u X. v))
24	19, 23	syl5ib 223	. . . . 5 |- ((R e. Top /\ S e. Top) -> (x e. {z | E.u e. R E.v e. S z = (u X. v)} -> x C_ (X X. Y)))
25	24	r19.21aiv 2175	. . . 4 |- ((R e. Top /\ S e. Top) -> A.x e. {z | E.u e. R E.v e. S z = (u X. v)}x C_ (X X. Y))
26		unissb 3208	. . . 4 |- (U.{z | E.u e. R E.v e. S z = (u X. v)} C_ (X X. Y) <-> A.x e. {z | E.u e. R E.v e. S z = (u X. v)}x C_ (X X. Y))
27	25, 26	sylibr 217	. . 3 |- ((R e. Top /\ S e. Top) -> U.{z | E.u e. R E.v e. S z = (u X. v)} C_ (X X. Y))
28	9	topopn 8871	. . . . . . 7 |- (R e. Top -> X e. R)
29	28	adantr 425	. . . . . 6 |- ((R e. Top /\ S e. Top) -> X e. R)
30	11	topopn 8871	. . . . . . 7 |- (S e. Top -> Y e. S)
31	30	adantl 424	. . . . . 6 |- ((R e. Top /\ S e. Top) -> Y e. S)
32		eqidd 1885	. . . . . 6 |- ((R e. Top /\ S e. Top) -> (X X. Y) = (X X. Y))
33		xpeq1 4016	. . . . . . . 8 |- (u = X -> (u X. v) = (X X. v))
34	33	eqeq2d 1895	. . . . . . 7 |- (u = X -> ((X X. Y) = (u X. v) <-> (X X. Y) = (X X. v)))
35		xpeq2 4017	. . . . . . . 8 |- (v = Y -> (X X. v) = (X X. Y))
36	35	eqeq2d 1895	. . . . . . 7 |- (v = Y -> ((X X. Y) = (X X. v) <-> (X X. Y) = (X X. Y)))
37	34, 36	rcla42ev 2385	. . . . . 6 |- ((X e. R /\ Y e. S /\ (X X. Y) = (X X. Y)) -> E.u e. R E.v e. S (X X. Y) = (u X. v))
38	29, 31, 32, 37	syl111anc 1100	. . . . 5 |- ((R e. Top /\ S e. Top) -> E.u e. R E.v e. S (X X. Y) = (u X. v))
39		xpexg 4095	. . . . . . 7 |- ((X e. _V /\ Y e. _V) -> (X X. Y) e. _V)
40		uniexg 3795	. . . . . . . 8 |- (R e. Top -> U.R e. _V)
41	40, 9	syl5eqel 1975	. . . . . . 7 |- (R e. Top -> X e. _V)
42		uniexg 3795	. . . . . . . 8 |- (S e. Top -> U.S e. _V)
43	42, 11	syl5eqel 1975	. . . . . . 7 |- (S e. Top -> Y e. _V)
44	39, 41, 43	syl2an 503	. . . . . 6 |- ((R e. Top /\ S e. Top) -> (X X. Y) e. _V)
45		eqeq1 1890	. . . . . . . 8 |- (z = (X X. Y) -> (z = (u X. v) <-> (X X. Y) = (u X. v)))
46	45	2rexbidv 2141	. . . . . . 7 |- (z = (X X. Y) -> (E.u e. R E.v e. S z = (u X. v) <-> E.u e. R E.v e. S (X X. Y) = (u X. v)))
47	46	elabg 2405	. . . . . 6 |- ((X X. Y) e. _V -> ((X X. Y) e. {z | E.u e. R E.v e. S z = (u X. v)} <-> E.u e. R E.v e. S (X X. Y) = (u X. v)))
48	44, 47	syl 12	. . . . 5 |- ((R e. Top /\ S e. Top) -> ((X X. Y) e. {z | E.u e. R E.v e. S z = (u X. v)} <-> E.u e. R E.v e. S (X X. Y) = (u X. v)))
49	38, 48	mpbird 213	. . . 4 |- ((R e. Top /\ S e. Top) -> (X X. Y) e. {z | E.u e. R E.v e. S z = (u X. v)})
50		elssuni 3206	. . . 4 |- ((X X. Y) e. {z | E.u e. R E.v e. S z = (u X. v)} -> (X X. Y) C_ U.{z | E.u e. R E.v e. S z = (u X. v)})
51	49, 50	syl 12	. . 3 |- ((R e. Top /\ S e. Top) -> (X X. Y) C_ U.{z | E.u e. R E.v e. S z = (u X. v)})
52	27, 51	eqssd 2633	. 2 |- ((R e. Top /\ S e. Top) -> U.{z | E.u e. R E.v e. S z = (u X. v)} = (X X. Y))
53	4, 7, 52	3eqtrd 1929	1 |- ((R e. Top /\ S e. Top) -> U.T = (X X. Y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  U.cuni 3177   X. cxp 3984  ` cfv 3998  (class class class)co 4884  Topctop 8857  Basesctb 8859  topGenctg 8860   X.t ctx 8930

This theorem is referenced by:  tx1cn 10223  tx2cn 10224  uptx 10226  txcn 10227  2txcn 10229  ptincpw 14912  extopgrp 14980  txunii 15910  txcnoprab 15911  txsubsp 15912  txcld 15914  cnresoprab 15915  cnopropabco 15917  cnoproprabco 15919  txmet 15925  phtpycom 16050  phtpycolem3 16053  phtpycolem4 16054  phtpycolem5 16055  pcohtpylem3 16082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931

Copyright terms: Public domain