Theorem cnmpt2nd 14703

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 6243	. . . . . 6 |- 2nd : _V -onto-> _V
2		fofn 5499	. . . . . 6 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 2nd Fn _V
4		ssv 3214	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5388	. . . . 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 5623	. . . 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 5599	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
10	9	mpteq2ia 4129	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 2nd |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 2nd ` z ) )
11		vex 2774	. . . . 5 |- x e. _V
12		vex 2774	. . . . 5 |- y e. _V
13	11, 12	op2ndd 6234	. . . 4 |- ( z = <. x , y >. -> ( 2nd ` z ) = y )
14	13	mpompt 6036	. . 3 |- ( z e. ( X X. Y ) |-> ( 2nd ` z ) ) = ( x e. X , y e. Y |-> y )
15	8, 10, 14	3eqtri 2229	. 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 14684	. . 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 2292	1 |- ( ph -> ( x e. X , y e. Y |-> y ) e. ( ( J tX K ) Cn K ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165   |-> cmpt 4104   X. cxp 4672   |` cres 4676   Fn wfn 5265   -onto->wfo 5268   ` cfv 5270  (class class class)co 5943   e. cmpo 5945   2ndc2nd 6224  TopOnctopon 14424   Cn ccn 14599   tX ctx 14666

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-map 6736  df-topgen 13034  df-top 14412  df-topon 14425  df-bases 14457  df-cn 14602  df-tx 14667

This theorem is referenced by:  cnmptcom  14712  txhmeo  14733  txswaphmeo  14735  divcnap  14979  cnrehmeocntop  15024