Theorem cnmpt1st 20338

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 6793	. . . . . 6 |- 1st : _V -onto-> _V
2		fofn 5779	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 1st Fn _V
4		ssv 3509	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5676	. . . . 5 |- ( ( 1st Fn _V /\ ( X X. Y ) C_ _V ) -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
6	3, 4, 5	mp2an 670	. . . 4 |- ( 1st |` ( X X. Y ) ) Fn ( X X. Y )
7		dffn5 5893	. . . 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	mpbi 208	. . 3 |- ( 1st |` ( X X. Y ) ) = ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) )
9		fvres 5862	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
10	9	mpteq2ia 4521	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 1st ` z ) )
11		vex 3109	. . . . 5 |- x e. _V
12		vex 3109	. . . . 5 |- y e. _V
13	11, 12	op1std 6783	. . . 4 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
14	13	mpt2mpt 6367	. . 3 |- ( z e. ( X X. Y ) |-> ( 1st ` z ) ) = ( x e. X , y e. Y |-> x )
15	8, 10, 14	3eqtri 2487	. 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 20279	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( J tX K ) Cn J ) )
19	16, 17, 18	syl2anc 659	. 2 |- ( ph -> ( 1st |` ( X X. Y ) ) e. ( ( J tX K ) Cn J ) )
20	15, 19	syl5eqelr 2547	1 |- ( ph -> ( x e. X , y e. Y |-> x ) e. ( ( J tX K ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 1823   _Vcvv 3106   C_ wss 3461   |-> cmpt 4497   X. cxp 4986   |` cres 4990   Fn wfn 5565   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270   |-> cmpt2 6272   1stc1st 6771  TopOnctopon 19565   Cn ccn 19895   tX ctx 20230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-topgen 14936  df-top 19569  df-bases 19571  df-topon 19572  df-cn 19898  df-tx 20232

This theorem is referenced by:  cnmptcom  20348  xkofvcn  20354  cnmptk2  20356  txhmeo  20473  txswaphmeo  20475  ptunhmeo  20478  xkohmeo  20485  tgpsubcn  20758  istgp2  20759  oppgtmd  20765  prdstmdd  20791  dvrcn  20855  divcn  21541  cnrehmeo  21622  htpycom  21645  htpyid  21646  htpyco1  21647  htpycc  21649  reparphti  21666  pcocn  21686  pcohtpylem  21688  pcopt  21691  pcopt2  21692  pcoass  21693  pcorevlem  21695  cxpcn  23290  vmcn  25810  dipcn  25834  mndpluscn  28146  cvxscon  28955  cvmlift2lem12  29026