Theorem cnmpt11f 21467

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

Hypotheses

Ref	Expression
cnmptid.j	|- ( ph -> J e. (TopOn ` X ) )
cnmpt11.a	|- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )
cnmpt11f.f	|- ( ph -> F e. ( K Cn L ) )

Assertion

Ref	Expression
cnmpt11f	|- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Distinct variable groups:   x, F   ph, x   x, J   x, X   x, K   x, L

Allowed substitution hint:   A( x)

Proof of Theorem cnmpt11f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptid.j	. 2 |- ( ph -> J e. (TopOn ` X ) )
2		cnmpt11.a	. 2 |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )
3		cntop2 21045	. . . 4 |- ( ( x e. X |-> A ) e. ( J Cn K ) -> K e. Top )
4	2, 3	syl 17	. . 3 |- ( ph -> K e. Top )
5		eqid 2622	. . . 4 |- U. K = U. K
6	5	toptopon 20722	. . 3 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
7	4, 6	sylib 208	. 2 |- ( ph -> K e. (TopOn ` U. K ) )
8		cnmpt11f.f	. . . . 5 |- ( ph -> F e. ( K Cn L ) )
9		eqid 2622	. . . . . 6 |- U. L = U. L
10	5, 9	cnf 21050	. . . . 5 |- ( F e. ( K Cn L ) -> F : U. K --> U. L )
11	8, 10	syl 17	. . . 4 |- ( ph -> F : U. K --> U. L )
12	11	feqmptd 6249	. . 3 |- ( ph -> F = ( y e. U. K |-> ( F ` y ) ) )
13	12, 8	eqeltrrd 2702	. 2 |- ( ph -> ( y e. U. K |-> ( F ` y ) ) e. ( K Cn L ) )
14		fveq2 6191	. 2 |- ( y = A -> ( F ` y ) = ( F ` A ) )
15	1, 2, 7, 13, 14	cnmpt11 21466	1 |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Syntax hints:   -> wi 4   e. wcel 1990   U.cuni 4436   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

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-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-map 7859  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  cnmpt12f  21469  tgpmulg  21897  prdstgpd  21928  pcorevcl  22825  pcorevlem  22826  logcn  24393  loglesqrt  24499  efrlim  24696  cvmliftlem8  31274  knoppcnlem10  32492  areacirclem2  33501  areacirclem4  33503