Theorem cnmpt1st 13082

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 6136	. . . . . 6 |- 1st : _V -onto-> _V
2		fofn 5422	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 1st Fn _V
4		ssv 3169	. . . . 5 |- ( X X. Y ) C_ _V
5		fnssres 5311	. . . . 5 |- ( ( 1st Fn _V /\ ( X X. Y ) C_ _V ) -> ( 1st |` ( X X. Y ) ) Fn ( X X. Y ) )
6	3, 4, 5	mp2an 424	. . . 4 |- ( 1st |` ( X X. Y ) ) Fn ( X X. Y )
7		dffn5im 5542	. . . 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 5520	. . . 4 |- ( z e. ( X X. Y ) -> ( ( 1st |` ( X X. Y ) ) ` z ) = ( 1st ` z ) )
10	9	mpteq2ia 4075	. . 3 |- ( z e. ( X X. Y ) |-> ( ( 1st |` ( X X. Y ) ) ` z ) ) = ( z e. ( X X. Y ) |-> ( 1st ` z ) )
11		vex 2733	. . . . 5 |- x e. _V
12		vex 2733	. . . . 5 |- y e. _V
13	11, 12	op1std 6127	. . . 4 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
14	13	mpompt 5945	. . 3 |- ( z e. ( X X. Y ) |-> ( 1st ` z ) ) = ( x e. X , y e. Y |-> x )
15	8, 10, 14	3eqtri 2195	. 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 13063	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( J tX K ) Cn J ) )
19	16, 17, 18	syl2anc 409	. 2 |- ( ph -> ( 1st |` ( X X. Y ) ) e. ( ( J tX K ) Cn J ) )
20	15, 19	eqeltrrid 2258	1 |- ( ph -> ( x e. X , y e. Y |-> x ) e. ( ( J tX K ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   |-> cmpt 4050   X. cxp 4609   |` cres 4613   Fn wfn 5193   -onto->wfo 5196   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1stc1st 6117  TopOnctopon 12802   Cn ccn 12979   tX ctx 13046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047

This theorem is referenced by:  cnmptcom  13092  txhmeo  13113  txswaphmeo  13115  divcnap  13349  cnrehmeocntop  13387