Theorem tx2cn 12478

Description: Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
tx2cn	|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )

Proof of Theorem tx2cn

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f2ndres 6066	. . 3 |- ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y
2	1	a1i 9	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y )
3		ffn 5280	. . . . . . . 8 |- ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y -> ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5547	. . . . . . . 8 |- ( ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) -> ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ ( ( 2nd |` ( X X. Y ) ) ` z ) e. w ) ) )
5	1, 3, 4	mp2b 8	. . . . . . 7 |- ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ ( ( 2nd |` ( X X. Y ) ) ` z ) e. w ) )
6		fvres 5453	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
7	6	eleq1d 2209	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> ( 2nd ` z ) e. w ) )
8		1st2nd2 6081	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp1st 6071	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
10		elxp6 6075	. . . . . . . . . . . 12 |- ( z e. ( X X. w ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. X /\ ( 2nd ` z ) e. w ) ) )
11		anass 399	. . . . . . . . . . . 12 |- ( ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) /\ ( 2nd ` z ) e. w ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. X /\ ( 2nd ` z ) e. w ) ) )
12	10, 11	bitr4i 186	. . . . . . . . . . 11 |- ( z e. ( X X. w ) <-> ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) /\ ( 2nd ` z ) e. w ) )
13	12	baib 905	. . . . . . . . . 10 |- ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
14	8, 9, 13	syl2anc 409	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
15	7, 14	bitr4d 190	. . . . . . . 8 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> z e. ( X X. w ) ) )
16	15	pm5.32i 450	. . . . . . 7 |- ( ( z e. ( X X. Y ) /\ ( ( 2nd |` ( X X. Y ) ) ` z ) e. w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) )
17	5, 16	bitri 183	. . . . . 6 |- ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) )
18		toponss 12232	. . . . . . . . . 10 |- ( ( S e. (TopOn ` Y ) /\ w e. S ) -> w C_ Y )
19	18	adantll 468	. . . . . . . . 9 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> w C_ Y )
20		xpss2 4658	. . . . . . . . 9 |- ( w C_ Y -> ( X X. w ) C_ ( X X. Y ) )
21	19, 20	syl 14	. . . . . . . 8 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( X X. w ) C_ ( X X. Y ) )
22	21	sseld 3101	. . . . . . 7 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( z e. ( X X. w ) -> z e. ( X X. Y ) ) )
23	22	pm4.71rd 392	. . . . . 6 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( z e. ( X X. w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) ) )
24	17, 23	bitr4id 198	. . . . 5 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> z e. ( X X. w ) ) )
25	24	eqrdv 2138	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) = ( X X. w ) )
26		toponmax 12231	. . . . . 6 |- ( R e. (TopOn ` X ) -> X e. R )
27		txopn 12473	. . . . . . 7 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ ( X e. R /\ w e. S ) ) -> ( X X. w ) e. ( R tX S ) )
28	27	expr 373	. . . . . 6 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ X e. R ) -> ( w e. S -> ( X X. w ) e. ( R tX S ) ) )
29	26, 28	mpidan 420	. . . . 5 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( w e. S -> ( X X. w ) e. ( R tX S ) ) )
30	29	imp 123	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( X X. w ) e. ( R tX S ) )
31	25, 30	eqeltrd 2217	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
32	31	ralrimiva 2508	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33		txtopon 12470	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
34		iscn 12405	. . 3 |- ( ( ( R tX S ) e. (TopOn ` ( X X. Y ) ) /\ S e. (TopOn ` Y ) ) -> ( ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) <-> ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y /\ A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) ) ) )
35	33, 34	sylancom 417	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) <-> ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y /\ A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) ) ) )
36	2, 32, 35	mpbir2and 929	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   C_ wss 3076   <.cop 3535   X. cxp 4545   `'ccnv 4546   |` cres 4549   "cima 4550   Fn wfn 5126   -->wf 5127   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045  TopOnctopon 12216   Cn ccn 12393   tX ctx 12460

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-topgen 12180  df-top 12204  df-topon 12217  df-bases 12249  df-cn 12396  df-tx 12461

This theorem is referenced by:  txcn  12483  cnmpt2nd  12497