Theorem cnmptid 14868

Description: The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypothesis

Ref	Expression
cnmptid.j	|- ( ph -> J e. (TopOn ` X ) )

Assertion

Ref	Expression
cnmptid	|- ( ph -> ( x e. X |-> x ) e. ( J Cn J ) )

Distinct variable groups:   ph, x   x, J   x, X

Proof of Theorem cnmptid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcom 1730	. . . . . 6 |- ( x = y <-> y = x )
2	1	opabbii 4127	. . . . 5 |- { <. x , y >. | x = y } = { <. x , y >. | y = x }
3		df-id 4358	. . . . 5 |- _I = { <. x , y >. | x = y }
4		mptv 4157	. . . . 5 |- ( x e. _V |-> x ) = { <. x , y >. | y = x }
5	2, 3, 4	3eqtr4i 2238	. . . 4 |- _I = ( x e. _V |-> x )
6	5	reseq1i 4974	. . 3 |- ( _I |` X ) = ( ( x e. _V |-> x ) |` X )
7		ssv 3223	. . . 4 |- X C_ _V
8		resmpt 5026	. . . 4 |- ( X C_ _V -> ( ( x e. _V |-> x ) |` X ) = ( x e. X |-> x ) )
9	7, 8	ax-mp 5	. . 3 |- ( ( x e. _V |-> x ) |` X ) = ( x e. X |-> x )
10	6, 9	eqtri 2228	. 2 |- ( _I |` X ) = ( x e. X |-> x )
11		cnmptid.j	. . 3 |- ( ph -> J e. (TopOn ` X ) )
12		idcn 14799	. . 3 |- ( J e. (TopOn ` X ) -> ( _I |` X ) e. ( J Cn J ) )
13	11, 12	syl 14	. 2 |- ( ph -> ( _I |` X ) e. ( J Cn J ) )
14	10, 13	eqeltrrid 2295	1 |- ( ph -> ( x e. X |-> x ) e. ( J Cn J ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   {copab 4120   |-> cmpt 4121   _I cid 4353   |` cres 4695   ` cfv 5290  (class class class)co 5967  TopOnctopon 14597   Cn ccn 14772

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-top 14585  df-topon 14598  df-cn 14775

This theorem is referenced by:  imasnopn  14886  expcn  15156