Theorem cncfcompt 40096

Description: Composition of continuous functions. A generalization of cncfmpt1f 22716 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncfcompt.bcn	|- ( ph -> ( x e. A |-> B ) e. ( A -cn-> C ) )
cncfcompt.f	|- ( ph -> F e. ( C -cn-> D ) )

Assertion

Ref	Expression
cncfcompt	|- ( ph -> ( x e. A |-> ( F ` B ) ) e. ( A -cn-> D ) )

Distinct variable groups:   x, A   x, C   x, D   x, F   ph, x

Allowed substitution hint:   B( x)

Proof of Theorem cncfcompt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfcompt.f	. . . . . 6 |- ( ph -> F e. ( C -cn-> D ) )
2		cncff 22696	. . . . . 6 |- ( F e. ( C -cn-> D ) -> F : C --> D )
3	1, 2	syl 17	. . . . 5 |- ( ph -> F : C --> D )
4	3	adantr 481	. . . 4 |- ( ( ph /\ x e. A ) -> F : C --> D )
5		cncfcompt.bcn	. . . . . 6 |- ( ph -> ( x e. A |-> B ) e. ( A -cn-> C ) )
6		cncff 22696	. . . . . 6 |- ( ( x e. A |-> B ) e. ( A -cn-> C ) -> ( x e. A |-> B ) : A --> C )
7	5, 6	syl 17	. . . . 5 |- ( ph -> ( x e. A |-> B ) : A --> C )
8	7	mptex2 6384	. . . 4 |- ( ( ph /\ x e. A ) -> B e. C )
9	4, 8	ffvelrnd 6360	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` B ) e. D )
10		eqid 2622	. . 3 |- ( x e. A |-> ( F ` B ) ) = ( x e. A |-> ( F ` B ) )
11	9, 10	fmptd 6385	. 2 |- ( ph -> ( x e. A |-> ( F ` B ) ) : A --> D )
12		cncfrss2 22695	. . . 4 |- ( F e. ( C -cn-> D ) -> D C_ CC )
13	1, 12	syl 17	. . 3 |- ( ph -> D C_ CC )
14		eqidd 2623	. . . . 5 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
15	3	feqmptd 6249	. . . . 5 |- ( ph -> F = ( y e. C |-> ( F ` y ) ) )
16		fveq2 6191	. . . . 5 |- ( y = B -> ( F ` y ) = ( F ` B ) )
17	8, 14, 15, 16	fmptco 6396	. . . 4 |- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )
18		ssid 3624	. . . . . . 7 |- CC C_ CC
19		cncfss 22702	. . . . . . 7 |- ( ( D C_ CC /\ CC C_ CC ) -> ( C -cn-> D ) C_ ( C -cn-> CC ) )
20	13, 18, 19	sylancl 694	. . . . . 6 |- ( ph -> ( C -cn-> D ) C_ ( C -cn-> CC ) )
21	20, 1	sseldd 3604	. . . . 5 |- ( ph -> F e. ( C -cn-> CC ) )
22	5, 21	cncfco 22710	. . . 4 |- ( ph -> ( F o. ( x e. A |-> B ) ) e. ( A -cn-> CC ) )
23	17, 22	eqeltrrd 2702	. . 3 |- ( ph -> ( x e. A |-> ( F ` B ) ) e. ( A -cn-> CC ) )
24		cncffvrn 22701	. . 3 |- ( ( D C_ CC /\ ( x e. A |-> ( F ` B ) ) e. ( A -cn-> CC ) ) -> ( ( x e. A |-> ( F ` B ) ) e. ( A -cn-> D ) <-> ( x e. A |-> ( F ` B ) ) : A --> D ) )
25	13, 23, 24	syl2anc 693	. 2 |- ( ph -> ( ( x e. A |-> ( F ` B ) ) e. ( A -cn-> D ) <-> ( x e. A |-> ( F ` B ) ) : A --> D ) )
26	11, 25	mpbird 247	1 |- ( ph -> ( x e. A |-> ( F ` B ) ) e. ( A -cn-> D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   -cn->ccncf 22679

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-po 5035  df-so 5036  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841  df-abs 13976  df-cncf 22681

This theorem is referenced by:  itgsbtaddcnst  40198  fourierdlem23  40347  fourierdlem83  40406  fourierdlem101  40424