Theorem cnmpt2nd 13660

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 6156	. . . . . 6 |- 2nd : _V -onto-> _V
2		fofn 5439	. . . . . 6 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 2nd Fn _V
4		ssv 3177	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5328	. . . . 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 5560	. . . 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 5538	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
10	9	mpteq2ia 4088	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 2nd |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 2nd ` z ) )
11		vex 2740	. . . . 5 |- x e. _V
12		vex 2740	. . . . 5 |- y e. _V
13	11, 12	op2ndd 6147	. . . 4 |- ( z = <. x , y >. -> ( 2nd ` z ) = y )
14	13	mpompt 5964	. . 3 |- ( z e. ( X X. Y ) |-> ( 2nd ` z ) ) = ( x e. X , y e. Y |-> y )
15	8, 10, 14	3eqtri 2202	. 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 13641	. . 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 2265	1 |- ( ph -> ( x e. X , y e. Y |-> y ) e. ( ( J tX K ) Cn K ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   |-> cmpt 4063   X. cxp 4623   |` cres 4627   Fn wfn 5210   -onto->wfo 5213   ` cfv 5215  (class class class)co 5872   e. cmpo 5874   2ndc2nd 6137  TopOnctopon 13379   Cn ccn 13556   tX ctx 13623

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-map 6647  df-topgen 12697  df-top 13367  df-topon 13380  df-bases 13412  df-cn 13559  df-tx 13624

This theorem is referenced by:  cnmptcom  13669  txhmeo  13690  txswaphmeo  13692  divcnap  13926  cnrehmeocntop  13964