Theorem fvmptelrn 5649

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
fvmptelrn.1	|- ( ph -> ( x e. A |-> B ) : A --> C )

Assertion

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

Distinct variable groups:   x, A   x, C

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

Proof of Theorem fvmptelrn

Step	Hyp	Ref	Expression
1		fvmptelrn.1	. . 3 |- ( ph -> ( x e. A |-> B ) : A --> C )
2		eqid 2170	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	fmpt 5646	. . 3 |- ( A. x e. A B e. C <-> ( x e. A |-> B ) : A --> C )
4	1, 3	sylibr 133	. 2 |- ( ph -> A. x e. A B e. C )
5	4	r19.21bi 2558	1 |- ( ( ph /\ x e. A ) -> B e. C )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   |-> cmpt 4050   -->wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  txcnp  13065  cnmpt1t  13079  cnmpt12  13081  dvmptclx  13474