Theorem fnmptd 14641

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 2550	. . 3 |- ( ph -> A. x e. A B e. V )
3		eqid 2177	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5344	. . 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 5308	. 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 1353   e. wcel 2148   A.wral 2455   |-> cmpt 4066   Fn wfn 5213

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-fun 5220  df-fn 5221

This theorem is referenced by: (None)