Theorem fnmptd 16403

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 2605	. . 3 |- ( ph -> A. x e. A B e. V )
3		eqid 2231	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5459	. . 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 5420	. 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 1397   e. wcel 2202   A.wral 2510   |-> cmpt 4150   Fn wfn 5321

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-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  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-fun 5328  df-fn 5329

This theorem is referenced by: (None)