Theorem fnmptd 13839

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 2543	. . 3 |- ( ph -> A. x e. A B e. V )
3		eqid 2170	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5324	. . 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 5288	. 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 1348   e. wcel 2141   A.wral 2448   |-> cmpt 4050   Fn wfn 5193

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-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  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-fun 5200  df-fn 5201

This theorem is referenced by: (None)