Theorem cnmpt1st 14962

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 6303	. . . . . 6 |- 1st : _V -onto-> _V
2		fofn 5550	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 1st Fn _V
4		ssv 3246	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5436	. . . . 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 5679	. . . 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 5651	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
10	9	mpteq2ia 4170	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 1st ` z ) )
11		vex 2802	. . . . 5 |- x e. _V
12		vex 2802	. . . . 5 |- y e. _V
13	11, 12	op1std 6294	. . . 4 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
14	13	mpompt 6096	. . 3 |- ( z e. ( X X. Y ) |-> ( 1st ` z ) ) = ( x e. X , y e. Y |-> x )
15	8, 10, 14	3eqtri 2254	. 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 14943	. . 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 2317	1 |- ( ph -> ( x e. X , y e. Y |-> x ) e. ( ( J tX K ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   |-> cmpt 4145   X. cxp 4717   |` cres 4721   Fn wfn 5313   -onto->wfo 5316   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   1stc1st 6284  TopOnctopon 14684   Cn ccn 14859   tX ctx 14926

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-topgen 13293  df-top 14672  df-topon 14685  df-bases 14717  df-cn 14862  df-tx 14927

This theorem is referenced by:  cnmptcom  14972  txhmeo  14993  txswaphmeo  14995  divcnap  15239  cnrehmeocntop  15284