Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exopcopn Unicode version

Theorem exopcopn 25675
 Description: For every neighborhood of in a product topology, there exist two open sets and of the component topologies so that is an open neighborhood of and a part of . (Use opelxp 4735 to have and .) (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
exopcopn.1
Assertion
Ref Expression
exopcopn
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem exopcopn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . . 4
2 exopcopn.1 . . . . . 6
3 txtop 17280 . . . . . . 7
433ad2ant1 976 . . . . . 6
52, 4syl5eqel 2380 . . . . 5
6 eqid 2296 . . . . . . . 8
76neii1 16859 . . . . . . 7
85, 1, 7syl2anc 642 . . . . . 6
9 opex 4253 . . . . . . . 8
109a1i 10 . . . . . . 7
11 elnei 16864 . . . . . . 7
125, 10, 1, 11syl3anc 1182 . . . . . 6
138, 12sseldd 3194 . . . . 5
146isneip 16858 . . . . 5
155, 13, 14syl2anc 642 . . . 4
161, 15mpbid 201 . . 3
1716simprd 449 . 2
18 simprr 733 . . . . . . 7
19 simprl 732 . . . . . . . . 9
2019, 2syl6eleq 2386 . . . . . . . 8
21 simpl1 958 . . . . . . . . 9
22 eltx 17279 . . . . . . . . 9
2321, 22syl 15 . . . . . . . 8
2420, 23mpbid 201 . . . . . . 7
25 eleq1 2356 . . . . . . . . . 10
2625anbi1d 685 . . . . . . . . 9
27262rexbidv 2599 . . . . . . . 8
2827rspcv 2893 . . . . . . 7
2918, 24, 28sylc 56 . . . . . 6
30 sstr2 3199 . . . . . . . . . 10
3130com12 27 . . . . . . . . 9
3231anim2d 548 . . . . . . . 8
3332reximdv 2667 . . . . . . 7
3433reximdv 2667 . . . . . 6
3529, 34syl5com 26 . . . . 5
3635anassrs 629 . . . 4
3736expimpd 586 . . 3
3837rexlimdva 2680 . 2
3917, 38mpd 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934   wceq 1632   wcel 1696  wral 2556  wrex 2557  cvv 2801   wss 3165  csn 3653  cop 3656  cuni 3843   cxp 4703  cfv 5271  (class class class)co 5874  ctop 16647  cnei 16850   ctx 17271 This theorem is referenced by:  limptlimpr2lem2  25678 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-topgen 13360  df-top 16652  df-bases 16654  df-nei 16851  df-tx 17273
 Copyright terms: Public domain W3C validator