Theorem cnmpt2nd 15154

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 6352	. . . . . 6 |- 2nd : _V -onto-> _V
2		fofn 5592	. . . . . 6 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 2nd Fn _V
4		ssv 3260	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5471	. . . . 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 5722	. . . 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 5694	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 2nd |` ( X X. Y ) ) ` z ) = ( 2nd ` z ) )
10	9	mpteq2ia 4196	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 2nd |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 2nd ` z ) )
11		vex 2816	. . . . 5 |- x e. _V
12		vex 2816	. . . . 5 |- y e. _V
13	11, 12	op2ndd 6343	. . . 4 |- ( z = <. x , y >. -> ( 2nd ` z ) = y )
14	13	mpompt 6145	. . 3 |- ( z e. ( X X. Y ) |-> ( 2nd ` z ) ) = ( x e. X , y e. Y |-> y )
15	8, 10, 14	3eqtri 2257	. 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 15135	. . 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 2320	1 |- ( ph -> ( x e. X , y e. Y |-> y ) e. ( ( J tX K ) Cn K ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   C_ wss 3211   |-> cmpt 4171   X. cxp 4747   |` cres 4751   Fn wfn 5347   -onto->wfo 5350   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   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:  cnmptcom  15163  txhmeo  15184  txswaphmeo  15186  divcnap  15430  cnrehmeocntop  15475