Theorem tx1cn 14437

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 6212	. . 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 5403	. . . . . . . 8 |- ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5677	. . . . . . . 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 5578	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
7	6	eleq1d 2262	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 1st |` ( X X. Y ) ) ` z ) e. w <-> ( 1st ` z ) e. w ) )
8		1st2nd2 6228	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp2nd 6219	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
10		elxp6 6222	. . . . . . . . . . . 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 920	. . . . . . . . . 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 14194	. . . . . . . . . 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 4769	. . . . . . . . 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 3178	. . . . . . 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 2191	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) = ( w X. Y ) )
27		toponmax 14193	. . . . . 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 14433	. . . . . 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 2270	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33	32	ralrimiva 2567	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. R ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
34		txtopon 14430	. . 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 14365	. . 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 946	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 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   <.cop 3621   X. cxp 4657   `'ccnv 4658   |` cres 4661   "cima 4662   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192  TopOnctopon 14178   Cn ccn 14353   tX ctx 14420

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-topgen 12871  df-top 14166  df-topon 14179  df-bases 14211  df-cn 14356  df-tx 14421

This theorem is referenced by:  txcn  14443  cnmpt1st  14456