Theorem cnmpt1st 14524

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 6215	. . . . . 6 |- 1st : _V -onto-> _V
2		fofn 5482	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 1st Fn _V
4		ssv 3205	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5371	. . . . 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 5606	. . . 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 5582	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
10	9	mpteq2ia 4119	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 1st ` z ) )
11		vex 2766	. . . . 5 |- x e. _V
12		vex 2766	. . . . 5 |- y e. _V
13	11, 12	op1std 6206	. . . 4 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
14	13	mpompt 6014	. . 3 |- ( z e. ( X X. Y ) |-> ( 1st ` z ) ) = ( x e. X , y e. Y |-> x )
15	8, 10, 14	3eqtri 2221	. 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 14505	. . 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 2284	1 |- ( ph -> ( x e. X , y e. Y |-> x ) e. ( ( J tX K ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   |-> cmpt 4094   X. cxp 4661   |` cres 4665   Fn wfn 5253   -onto->wfo 5256   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   1stc1st 6196  TopOnctopon 14246   Cn ccn 14421   tX ctx 14488

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-topgen 12931  df-top 14234  df-topon 14247  df-bases 14279  df-cn 14424  df-tx 14489

This theorem is referenced by:  cnmptcom  14534  txhmeo  14555  txswaphmeo  14557  divcnap  14801  cnrehmeocntop  14846