Theorem tx2cn 13064

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 6139	. . 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 5347	. . . . . . . 8 |- ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) --> Y -> ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) )
4		elpreima 5615	. . . . . . . 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 5520	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
7	6	eleq1d 2239	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> ( 2nd ` z ) e. w ) )
8		1st2nd2 6154	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
9		xp1st 6144	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
10		elxp6 6148	. . . . . . . . . . . 12 |- ( z e. ( X X. w ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. X /\ ( 2nd ` z ) e. w ) ) )
11		anass 399	. . . . . . . . . . . 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 186	. . . . . . . . . . 11 |- ( z e. ( X X. w ) <-> ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) /\ ( 2nd ` z ) e. w ) )
13	12	baib 914	. . . . . . . . . 10 |- ( ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( 1st ` z ) e. X ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
14	8, 9, 13	syl2anc 409	. . . . . . . . 9 |- ( z e. ( X X. Y ) -> ( z e. ( X X. w ) <-> ( 2nd ` z ) e. w ) )
15	7, 14	bitr4d 190	. . . . . . . 8 |- ( z e. ( X X. Y ) -> ( ( ( 2nd |` ( X X. Y ) ) ` z ) e. w <-> z e. ( X X. w ) ) )
16	15	pm5.32i 451	. . . . . . 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 183	. . . . . 6 |- ( z e. ( `' ( 2nd |` ( X X. Y ) ) " w ) <-> ( z e. ( X X. Y ) /\ z e. ( X X. w ) ) )
18		toponss 12818	. . . . . . . . . 10 |- ( ( S e. (TopOn ` Y ) /\ w e. S ) -> w C_ Y )
19	18	adantll 473	. . . . . . . . 9 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> w C_ Y )
20		xpss2 4722	. . . . . . . . 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 3146	. . . . . . 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 392	. . . . . 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 198	. . . . 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 2168	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) = ( X X. w ) )
26		toponmax 12817	. . . . . 6 |- ( R e. (TopOn ` X ) -> X e. R )
27		txopn 13059	. . . . . . 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 373	. . . . . 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 421	. . . . 5 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( w e. S -> ( X X. w ) e. ( R tX S ) ) )
30	29	imp 123	. . . 4 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( X X. w ) e. ( R tX S ) )
31	25, 30	eqeltrd 2247	. . 3 |- ( ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) /\ w e. S ) -> ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
32	31	ralrimiva 2543	. 2 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> A. w e. S ( `' ( 2nd |` ( X X. Y ) ) " w ) e. ( R tX S ) )
33		txtopon 13056	. . 3 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
34		iscn 12991	. . 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 418	. 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 939	1 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   <.cop 3586   X. cxp 4609   `'ccnv 4610   |` cres 4613   "cima 4614   Fn wfn 5193   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118  TopOnctopon 12802   Cn ccn 12979   tX ctx 13046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047

This theorem is referenced by:  txcn  13069  cnmpt2nd  13083