Theorem tx1cn 14943

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

Assertion

Ref	Expression
tx1cn	|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )

Proof of Theorem tx1cn

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

Step	Hyp	Ref	Expression
1		f1stres 6305	. . 3 |- ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X
2	1	a1i 9	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X )
3		ffn 5473	. . . . . . . 8 |- ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5754	. . . . . . . 8 |- ( ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) -> ( z e. ( `' ( 1st |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ ( ( 1st |` ( X X. Y ) ) ` z ) e. w ) ) )
5	1, 3, 4	mp2b 8	. . . . . . 7 |- ( z e. ( `' ( 1st |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ ( ( 1st |` ( X X. Y ) ) ` z ) e. w ) )
6		fvres 5651	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
7	6	eleq1d 2298	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 1st |` ( X X. Y ) ) ` z ) e. w <-> ( 1st ` z ) e. w ) )
8		1st2nd2 6321	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp2nd 6312	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
10		elxp6 6315	. . . . . . . . . . . 12 |- ( z e. ( w X. Y ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. w /\ ( 2nd ` z ) e. Y ) ) )
11		anass 401	. . . . . . . . . . . 12 |- ( ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. w ) /\ ( 2nd ` z ) e. Y ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. w /\ ( 2nd ` z ) e. Y ) ) )
12		an32 562	. . . . . . . . . . . 12 |- ( ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. w ) /\ ( 2nd ` z ) e. Y ) <-> ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 2nd ` z ) e. Y ) /\ ( 1st ` z ) e. w ) )
13	10, 11, 12	3bitr2i 208	. . . . . . . . . . 11 |- ( z e. ( w X. Y ) <-> ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 2nd ` z ) e. Y ) /\ ( 1st ` z ) e. w ) )
14	13	baib 924	. . . . . . . . . 10 |- ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 2nd ` z ) e. Y ) -> ( z e. ( w X. Y ) <-> ( 1st ` z ) e. w ) )
15	8, 9, 14	syl2anc 411	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( z e. ( w X. Y ) <-> ( 1st ` z ) e. w ) )
16	7, 15	bitr4d 191	. . . . . . . 8 |- ( z e. ( X X. Y ) -> ( ( ( 1st |` ( X X. Y ) ) ` z ) e. w <-> z e. ( w X. Y ) ) )
17	16	pm5.32i 454	. . . . . . 7 |- ( ( z e. ( X X. Y ) /\ ( ( 1st |` ( X X. Y ) ) ` z ) e. w ) <-> ( z e. ( X X. Y ) /\ z e. ( w X. Y ) ) )
18	5, 17	bitri 184	. . . . . 6 |- ( z e. ( `' ( 1st |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ z e. ( w X. Y ) ) )
19		toponss 14700	. . . . . . . . . 10 |- ( ( R e. (TopOn ` X ) /\ w e. R ) -> w C_ X )
20	19	adantlr 477	. . . . . . . . 9 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> w C_ X )
21		xpss1 4829	. . . . . . . . 9 |- ( w C_ X -> ( w X. Y ) C_ ( X X. Y ) )
22	20, 21	syl 14	. . . . . . . 8 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( w X. Y ) C_ ( X X. Y ) )
23	22	sseld 3223	. . . . . . 7 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( z e. ( w X. Y ) -> z e. ( X X. Y ) ) )
24	23	pm4.71rd 394	. . . . . 6 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( z e. ( w X. Y ) <-> ( z e. ( X X. Y ) /\ z e. ( w X. Y ) ) ) )
25	18, 24	bitr4id 199	. . . . 5 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( z e. ( `' ( 1st |` ( X X. Y ) ) " w ) <-> z e. ( w X. Y ) ) )
26	25	eqrdv 2227	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) = ( w X. Y ) )
27		toponmax 14699	. . . . . 6 |- ( S e. (TopOn ` Y ) -> Y e. S )
28	27	ad2antlr 489	. . . . 5 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> Y e. S )
29		txopn 14939	. . . . . 6 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ ( w e. R /\ Y e. S ) ) -> ( w X. Y ) e. ( R tX S ) )
30	29	anassrs 400	. . . . 5 |- ( ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) /\ Y e. S ) -> ( w X. Y ) e. ( R tX S ) )
31	28, 30	mpdan 421	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( w X. Y ) e. ( R tX S ) )
32	26, 31	eqeltrd 2306	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33	32	ralrimiva 2603	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. R ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
34		txtopon 14936	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
35		simpl 109	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> R e. (TopOn ` X ) )
36		iscn 14871	. . 3 |- ( ( ( R tX S ) e. (TopOn ` ( X X. Y ) ) /\ R e. (TopOn ` X ) ) -> ( ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) <-> ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X /\ A. w e. R ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) ) ) )
37	34, 35, 36	syl2anc 411	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) <-> ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X /\ A. w e. R ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) ) ) )
38	2, 33, 37	mpbir2and 950	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   <.cop 3669   X. cxp 4717   `'ccnv 4718   |` cres 4721   "cima 4722   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285  TopOnctopon 14684   Cn ccn 14859   tX ctx 14926

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-topgen 13293  df-top 14672  df-topon 14685  df-bases 14717  df-cn 14862  df-tx 14927

This theorem is referenced by:  txcn  14949  cnmpt1st  14962