Theorem cnmpt2nd 15012

Description: The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmpt21.j	|- ( ph -> J e. (TopOn ` X ) )
cnmpt21.k	|- ( ph -> K e. (TopOn ` Y ) )

Assertion

Ref	Expression
cnmpt2nd	|- ( ph -> ( x e. X , y e. Y |-> y ) e. ( ( J tX K ) Cn K ) )

Distinct variable groups:   x, y, ph   x, X, y   x, Y, y

Allowed substitution hints:   J( x, y)   K( x, y)

Proof of Theorem cnmpt2nd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 6320	. . . . . 6 |- 2nd : _V -onto-> _V
2		fofn 5561	. . . . . 6 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 2nd Fn _V
4		ssv 3249	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5445	. . . . 5 |- ( ( 2nd Fn _V /\ ( X X. Y ) C_ _V ) -> ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) )
6	3, 4, 5	mp2an 426	. . . 4 |- ( 2nd |` ( X X. Y ) ) Fn ( X X. Y )
7		dffn5im 5691	. . . 4 |- ( ( 2nd |` ( X X. Y ) ) Fn ( X X. Y ) -> ( 2nd |` ( X X. Y ) ) = ( z e. ( X X. Y ) |-> ( ( 2nd |` ( X X. Y ) ) ` z ) ) )
8	6, 7	ax-mp 5	. . 3 |- ( 2nd |` ( X X. Y ) ) = ( z e. ( X X. Y ) |-> ( ( 2nd |` ( X X. Y ) ) ` z ) )
9		fvres 5663	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
10	9	mpteq2ia 4175	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 2nd |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 2nd ` z ) )
11		vex 2805	. . . . 5 |- x e. _V
12		vex 2805	. . . . 5 |- y e. _V
13	11, 12	op2ndd 6311	. . . 4 |- ( z = <. x , y >. -> ( 2nd ` z ) = y )
14	13	mpompt 6112	. . 3 |- ( z e. ( X X. Y ) |-> ( 2nd ` z ) ) = ( x e. X , y e. Y |-> y )
15	8, 10, 14	3eqtri 2256	. 2 |- ( 2nd |` ( X X. Y ) ) = ( x e. X , y e. Y |-> y )
16		cnmpt21.j	. . 3 |- ( ph -> J e. (TopOn ` X ) )
17		cnmpt21.k	. . 3 |- ( ph -> K e. (TopOn ` Y ) )
18		tx2cn 14993	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( J tX K ) Cn K ) )
19	16, 17, 18	syl2anc 411	. 2 |- ( ph -> ( 2nd |` ( X X. Y ) ) e. ( ( J tX K ) Cn K ) )
20	15, 19	eqeltrrid 2319	1 |- ( ph -> ( x e. X , y e. Y |-> y ) e. ( ( J tX K ) Cn K ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   |-> cmpt 4150   X. cxp 4723   |` cres 4727   Fn wfn 5321   -onto->wfo 5324   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   2ndc2nd 6301  TopOnctopon 14733   Cn ccn 14908   tX ctx 14975

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-topgen 13342  df-top 14721  df-topon 14734  df-bases 14766  df-cn 14911  df-tx 14976

This theorem is referenced by:  cnmptcom  15021  txhmeo  15042  txswaphmeo  15044  divcnap  15288  cnrehmeocntop  15333