Theorem tx1cn 15134

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 6353	. . 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 5508	. . . . . . . 8 |- ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) --> X -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5797	. . . . . . . 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 5694	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
7	6	eleq1d 2301	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 1st |` ( X X. Y ) ) ` z ) e. w <-> ( 1st ` z ) e. w ) )
8		1st2nd2 6369	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp2nd 6360	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
10		elxp6 6363	. . . . . . . . . . . 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 564	. . . . . . . . . . . 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 927	. . . . . . . . . 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 14891	. . . . . . . . . 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 4860	. . . . . . . . 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 3237	. . . . . . 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 2230	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) = ( w X. Y ) )
27		toponmax 14890	. . . . . 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 15130	. . . . . 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 2309	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. R ) -> ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33	32	ralrimiva 2615	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. R ( `' ( 1st |` ( X X. Y ) ) " w ) e. ( R tX S ) )
34		txtopon 15127	. . 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 15062	. . 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 953	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 1398   e. wcel 2203   A.wral 2520   C_ wss 3211   <.cop 3692   X. cxp 4747   `'ccnv 4748   |` cres 4751   "cima 4752   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   1stc1st 6332   2ndc2nd 6333  TopOnctopon 14875   Cn ccn 15050   tX ctx 15117

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-topgen 13473  df-top 14863  df-topon 14876  df-bases 14908  df-cn 15053  df-tx 15118

This theorem is referenced by:  txcn  15140  cnmpt1st  15153