Theorem fvmptelcdm 5800

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 2231	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	fmpt 5797	. . 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 2620	1 |- ( ( ph /\ x e. A ) -> B e. C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   A.wral 2510   |-> cmpt 4150   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by:  txcnp  14994  cnmpt1t  15008  cnmpt12  15010  divcncfap  15337  maxcncf  15338  mincncf  15339  dvmptclx  15441