Theorem cncfmptss 39819

Description: A continuous complex function restricted to a subset is continuous, using "map to" notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
cncfmptss.1	|- F/_ x F
cncfmptss.2	|- ( ph -> F e. ( A -cn-> B ) )
cncfmptss.3	|- ( ph -> C C_ A )

Assertion

Ref	Expression
cncfmptss	|- ( ph -> ( x e. C |-> ( F ` x ) ) e. ( C -cn-> B ) )

Distinct variable group:   x, C

Allowed substitution hints:   ph( x)   A( x)   B( x)   F( x)

Proof of Theorem cncfmptss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfmptss.3	. . . 4 |- ( ph -> C C_ A )
2	1	resmptd 5452	. . 3 |- ( ph -> ( ( y e. A |-> ( F ` y ) ) |` C ) = ( y e. C |-> ( F ` y ) ) )
3		cncfmptss.2	. . . . . 6 |- ( ph -> F e. ( A -cn-> B ) )
4		cncff 22696	. . . . . 6 |- ( F e. ( A -cn-> B ) -> F : A --> B )
5	3, 4	syl 17	. . . . 5 |- ( ph -> F : A --> B )
6	5	feqmptd 6249	. . . 4 |- ( ph -> F = ( y e. A |-> ( F ` y ) ) )
7	6	reseq1d 5395	. . 3 |- ( ph -> ( F |` C ) = ( ( y e. A |-> ( F ` y ) ) |` C ) )
8		nfcv 2764	. . . . . 6 |- F/_ y F
9		nfcv 2764	. . . . . 6 |- F/_ y x
10	8, 9	nffv 6198	. . . . 5 |- F/_ y ( F ` x )
11		cncfmptss.1	. . . . . 6 |- F/_ x F
12		nfcv 2764	. . . . . 6 |- F/_ x y
13	11, 12	nffv 6198	. . . . 5 |- F/_ x ( F ` y )
14		fveq2 6191	. . . . 5 |- ( x = y -> ( F ` x ) = ( F ` y ) )
15	10, 13, 14	cbvmpt 4749	. . . 4 |- ( x e. C |-> ( F ` x ) ) = ( y e. C |-> ( F ` y ) )
16	15	a1i 11	. . 3 |- ( ph -> ( x e. C |-> ( F ` x ) ) = ( y e. C |-> ( F ` y ) ) )
17	2, 7, 16	3eqtr4rd 2667	. 2 |- ( ph -> ( x e. C |-> ( F ` x ) ) = ( F |` C ) )
18		rescncf 22700	. . 3 |- ( C C_ A -> ( F e. ( A -cn-> B ) -> ( F |` C ) e. ( C -cn-> B ) ) )
19	1, 3, 18	sylc 65	. 2 |- ( ph -> ( F |` C ) e. ( C -cn-> B ) )
20	17, 19	eqeltrd 2701	1 |- ( ph -> ( x e. C |-> ( F ` x ) ) e. ( C -cn-> B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   C_ wss 3574   |-> cmpt 4729   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   -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

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-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-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-cncf 22681

This theorem is referenced by:  cncfmptssg  40083  itgsin0pilem1  40165  ibliccsinexp  40166  itgsinexplem1  40169  itgsinexp  40170