Theorem tx2cn 14449

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 6215	. . 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 5404	. . . . . . . 8 |- ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y -> ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5678	. . . . . . . 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 5579	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
7	6	eleq1d 2262	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> ( 2nd ` z ) e. w ) )
8		1st2nd2 6230	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp1st 6220	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
10		elxp6 6224	. . . . . . . . . . . 12 |- ( z e. ( X X. w ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. X /\ ( 2nd ` z ) e. w ) ) )
11		anass 401	. . . . . . . . . . . 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 187	. . . . . . . . . . 11 |- ( z e. ( X X. w ) <-> ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) /\ ( 2nd ` z ) e. w ) )
13	12	baib 920	. . . . . . . . . 10 |- ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
14	8, 9, 13	syl2anc 411	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
15	7, 14	bitr4d 191	. . . . . . . 8 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> z e. ( X X. w ) ) )
16	15	pm5.32i 454	. . . . . . 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 184	. . . . . 6 |- ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) )
18		toponss 14205	. . . . . . . . . 10 |- ( ( S e. (TopOn ` Y ) /\ w e. S ) -> w C_ Y )
19	18	adantll 476	. . . . . . . . 9 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> w C_ Y )
20		xpss2 4771	. . . . . . . . 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 3179	. . . . . . 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 394	. . . . . 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 199	. . . . 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 2191	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) = ( X X. w ) )
26		toponmax 14204	. . . . . 6 |- ( R e. (TopOn ` X ) -> X e. R )
27		txopn 14444	. . . . . . 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 375	. . . . . 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 423	. . . . 5 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( w e. S -> ( X X. w ) e. ( R tX S ) ) )
30	29	imp 124	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( X X. w ) e. ( R tX S ) )
31	25, 30	eqeltrd 2270	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
32	31	ralrimiva 2567	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33		txtopon 14441	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
34		iscn 14376	. . 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 420	. 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 946	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3154   <.cop 3622   X. cxp 4658   `'ccnv 4659   |` cres 4662   "cima 4663   Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5919   1stc1st 6193   2ndc2nd 6194  TopOnctopon 14189   Cn ccn 14364   tX ctx 14431

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-topgen 12874  df-top 14177  df-topon 14190  df-bases 14222  df-cn 14367  df-tx 14432

This theorem is referenced by:  txcn  14454  cnmpt2nd  14468