Theorem fnmptd 13686

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 2539	. . 3 |- ( ph -> A. x e. A B e. V )
3		eqid 2165	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5314	. . 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 5278	. 2 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 166	1 |- ( ph -> F Fn A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   |-> cmpt 4043   Fn wfn 5183

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by: (None)