Theorem fvmptelcdm 5788

Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
fvmptelcdm.1	|- ( ph -> ( x e. A |-> B ) : A --> C )

Assertion

Ref	Expression
fvmptelcdm	|- ( ( ph /\ x e. A ) -> B e. C )

Distinct variable groups:   x, A   x, C

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

Proof of Theorem fvmptelcdm

Step	Hyp	Ref	Expression
1		fvmptelcdm.1	. . 3 |- ( ph -> ( x e. A |-> B ) : A --> C )
2		eqid 2229	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	fmpt 5785	. . 3 |- ( A. x e. A B e. C <-> ( x e. A |-> B ) : A --> C )
4	1, 3	sylibr 134	. 2 |- ( ph -> A. x e. A B e. C )
5	4	r19.21bi 2618	1 |- ( ( ph /\ x e. A ) -> B e. C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508   |-> cmpt 4145   -->wf 5314

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by:  txcnp  14945  cnmpt1t  14959  cnmpt12  14961  divcncfap  15288  maxcncf  15289  mincncf  15290  dvmptclx  15392