Theorem cncnp2 21085

Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Hypotheses

Ref	Expression
cncnp.1	|- X = U. J
cncnp.2	|- Y = U. K

Assertion

Ref	Expression
cncnp2	|- ( X =/= (/) -> ( F e. ( J Cn K ) <-> A. x e. X F e. ( ( J CnP K ) ` x ) ) )

Distinct variable groups:   x, F   x, J   x, K   x, X   x, Y

Proof of Theorem cncnp2

Step	Hyp	Ref	Expression
1		cntop1 21044	. . . . 5 |- ( F e. ( J Cn K ) -> J e. Top )
2		cncnp.1	. . . . . 6 |- X = U. J
3	2	toptopon 20722	. . . . 5 |- ( J e. Top <-> J e. (TopOn ` X ) )
4	1, 3	sylib 208	. . . 4 |- ( F e. ( J Cn K ) -> J e. (TopOn ` X ) )
5		cntop2 21045	. . . . 5 |- ( F e. ( J Cn K ) -> K e. Top )
6		cncnp.2	. . . . . 6 |- Y = U. K
7	6	toptopon 20722	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` Y ) )
8	5, 7	sylib 208	. . . 4 |- ( F e. ( J Cn K ) -> K e. (TopOn ` Y ) )
9	2, 6	cnf 21050	. . . 4 |- ( F e. ( J Cn K ) -> F : X --> Y )
10	4, 8, 9	jca31 557	. . 3 |- ( F e. ( J Cn K ) -> ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) )
11	10	adantl 482	. 2 |- ( ( X =/= (/) /\ F e. ( J Cn K ) ) -> ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) )
12		r19.2z 4060	. . 3 |- ( ( X =/= (/) /\ A. x e. X F e. ( ( J CnP K ) ` x ) ) -> E. x e. X F e. ( ( J CnP K ) ` x ) )
13		cnptop1 21046	. . . . . 6 |- ( F e. ( ( J CnP K ) ` x ) -> J e. Top )
14	13, 3	sylib 208	. . . . 5 |- ( F e. ( ( J CnP K ) ` x ) -> J e. (TopOn ` X ) )
15		cnptop2 21047	. . . . . 6 |- ( F e. ( ( J CnP K ) ` x ) -> K e. Top )
16	15, 7	sylib 208	. . . . 5 |- ( F e. ( ( J CnP K ) ` x ) -> K e. (TopOn ` Y ) )
17	2, 6	cnpf 21051	. . . . 5 |- ( F e. ( ( J CnP K ) ` x ) -> F : X --> Y )
18	14, 16, 17	jca31 557	. . . 4 |- ( F e. ( ( J CnP K ) ` x ) -> ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) )
19	18	rexlimivw 3029	. . 3 |- ( E. x e. X F e. ( ( J CnP K ) ` x ) -> ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) )
20	12, 19	syl 17	. 2 |- ( ( X =/= (/) /\ A. x e. X F e. ( ( J CnP K ) ` x ) ) -> ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) )
21		cncnp 21084	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. X F e. ( ( J CnP K ) ` x ) ) ) )
22	21	baibd 948	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( F e. ( J Cn K ) <-> A. x e. X F e. ( ( J CnP K ) ` x ) ) )
23	11, 20, 22	pm5.21nd 941	1 |- ( X =/= (/) -> ( F e. ( J Cn K ) <-> A. x e. X F e. ( ( J CnP K ) ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   U.cuni 4436   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   CnP ccnp 21029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-top 20699  df-topon 20716  df-cn 21031  df-cnp 21032

This theorem is referenced by: (None)