Theorem tx1cn 15370

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 6393	. . 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 5533	. . . . . . . 8 |- ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5828	. . . . . . . 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 5719	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
7	6	eleq1d 2307	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 1st |` ( X X. Y ) ) ` z ) e. w <-> ( 1st ` z ) e. w ) )
8		1st2nd2 6409	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp2nd 6400	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
10		elxp6 6403	. . . . . . . . . . . 12 |- ( z e. ( w X. Y ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. w /\ ( 2nd ` z ) e. Y ) ) )
11		anass 405	. . . . . . . . . . . 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 568	. . . . . . . . . . . 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 931	. . . . . . . . . 10 |- ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 2nd ` z ) e. Y ) -> ( z e. ( w X. Y ) <-> ( 1st ` z ) e. w ) )
15	8, 9, 14	syl2anc 415	. . . . . . . . 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 458	. . . . . . 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 15127	. . . . . . . . . 10 |- ( ( R e. (TopOn ` X ) /\ w e. R ) -> w C_ X )
20	19	adantlr 481	. . . . . . . . 9 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> w C_ X )
21		xpss1 4885	. . . . . . . . 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 3247	. . . . . . 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 398	. . . . . 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 2236	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) = ( w X. Y ) )
27		toponmax 15126	. . . . . 6 |- ( S e. (TopOn ` Y ) -> Y e. S )
28	27	ad2antlr 493	. . . . 5 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> Y e. S )
29		txopn 15366	. . . . . 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 404	. . . . 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 425	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( w X. Y ) e. ( R tX S ) )
32	26, 31	eqeltrd 2315	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33	32	ralrimiva 2623	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. R ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
34		txtopon 15363	. . 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 15298	. . 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 415	. 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 957	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   <.cop 3712   X. cxp 4772   `'ccnv 4773   |` cres 4776   "cima 4777   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373  TopOnctopon 15111   Cn ccn 15286   tX ctx 15353

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-topgen 13614  df-top 15099  df-topon 15112  df-bases 15144  df-cn 15289  df-tx 15354

This theorem is used by:  txcn  15376  cnmpt1st  15389