Theorem tx2cn 15294

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 6384	. . 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 5528	. . . . . . . 8 |- ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y -> ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5819	. . . . . . . 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 5714	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
7	6	eleq1d 2307	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> ( 2nd ` z ) e. w ) )
8		1st2nd2 6399	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp1st 6389	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
10		elxp6 6393	. . . . . . . . . . . 12 |- ( z e. ( X X. w ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. X /\ ( 2nd ` z ) e. w ) ) )
11		anass 405	. . . . . . . . . . . 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 931	. . . . . . . . . 10 |- ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
14	8, 9, 13	syl2anc 415	. . . . . . . . 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 458	. . . . . . 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 15050	. . . . . . . . . 10 |- ( ( S e. (TopOn ` Y ) /\ w e. S ) -> w C_ Y )
19	18	adantll 480	. . . . . . . . 9 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> w C_ Y )
20		xpss2 4881	. . . . . . . . 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 3247	. . . . . . 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 398	. . . . . 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 2236	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) = ( X X. w ) )
26		toponmax 15049	. . . . . 6 |- ( R e. (TopOn ` X ) -> X e. R )
27		txopn 15289	. . . . . . 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 427	. . . . 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 2315	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
32	31	ralrimiva 2623	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33		txtopon 15286	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
34		iscn 15221	. . 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 424	. 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 957	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 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   <.cop 3708   X. cxp 4767   `'ccnv 4768   |` cres 4771   "cima 4772   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363  TopOnctopon 15034   Cn ccn 15209   tX ctx 15276

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-topgen 13591  df-top 15022  df-topon 15035  df-bases 15067  df-cn 15212  df-tx 15277

This theorem is referenced by:  txcn  15299  cnmpt2nd  15313