Theorem fnmptd 15296

Description: The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)

Hypotheses

Ref	Expression
fnmptd.def	|- ( ph -> F = ( x e. A |-> B ) )
fnmptd.ex	|- ( ( ph /\ x e. A ) -> B e. V )

Assertion

Ref	Expression
fnmptd	|- ( ph -> F Fn A )

Distinct variable groups:   x, A   ph, x

Allowed substitution hints:   B( x)   F( x)   V( x)

Proof of Theorem fnmptd

Step	Hyp	Ref	Expression
1		fnmptd.ex	. . . 4 |- ( ( ph /\ x e. A ) -> B e. V )
2	1	ralrimiva 2567	. . 3 |- ( ph -> A. x e. A B e. V )
3		eqid 2193	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5380	. . 3 |- ( A. x e. A B e. V -> ( x e. A |-> B ) Fn A )
5	2, 4	syl 14	. 2 |- ( ph -> ( x e. A |-> B ) Fn A )
6		fnmptd.def	. . 3 |- ( ph -> F = ( x e. A |-> B ) )
7	6	fneq1d 5344	. 2 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 167	1 |- ( ph -> F Fn A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   |-> cmpt 4090   Fn wfn 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-fun 5256  df-fn 5257

This theorem is referenced by: (None)