Theorem cnmpt1st 15170

Description: The projection onto the first 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
cnmpt1st	|- ( ph -> ( x e. X , y e. Y |-> x ) e. ( ( J tX K ) Cn J ) )

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

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

Proof of Theorem cnmpt1st

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo1st 6353	. . . . . 6 |- 1st : _V -onto-> _V
2		fofn 5594	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 1st Fn _V
4		ssv 3262	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5473	. . . . 5 |- ( ( 1st Fn _V /\ ( X X. Y ) C_ _V ) -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
6	3, 4, 5	mp2an 426	. . . 4 |- ( 1st |` ( X X. Y ) ) Fn ( X X. Y )
7		dffn5im 5724	. . . 4 |- ( ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) -> ( 1st |` ( X X. Y ) ) = ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) ) )
8	6, 7	ax-mp 5	. . 3 |- ( 1st |` ( X X. Y ) ) = ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) )
9		fvres 5696	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
10	9	mpteq2ia 4198	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 1st ` z ) )
11		vex 2818	. . . . 5 |- x e. _V
12		vex 2818	. . . . 5 |- y e. _V
13	11, 12	op1std 6344	. . . 4 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
14	13	mpompt 6147	. . 3 |- ( z e. ( X X. Y ) |-> ( 1st ` z ) ) = ( x e. X , y e. Y |-> x )
15	8, 10, 14	3eqtri 2259	. 2 |- ( 1st |` ( X X. Y ) ) = ( x e. X , y e. Y |-> x )
16		cnmpt21.j	. . 3 |- ( ph -> J e. (TopOn ` X ) )
17		cnmpt21.k	. . 3 |- ( ph -> K e. (TopOn ` Y ) )
18		tx1cn 15151	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( J tX K ) Cn J ) )
19	16, 17, 18	syl2anc 411	. 2 |- ( ph -> ( 1st |` ( X X. Y ) ) e. ( ( J tX K ) Cn J ) )
20	15, 19	eqeltrrid 2322	1 |- ( ph -> ( x e. X , y e. Y |-> x ) e. ( ( J tX K ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   |-> cmpt 4173   X. cxp 4749   |` cres 4753   Fn wfn 5349   -onto->wfo 5352   ` cfv 5354  (class class class)co 6052   e. cmpo 6054   1stc1st 6334  TopOnctopon 14892   Cn ccn 15067   tX ctx 15134

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-topgen 13490  df-top 14880  df-topon 14893  df-bases 14925  df-cn 15070  df-tx 15135

This theorem is referenced by:  cnmptcom  15180  txhmeo  15201  txswaphmeo  15203  divcnap  15447  cnrehmeocntop  15492