Theorem funmptd 16575

Description: The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5389, then prove funmptd 16575 from it, and then prove funmpt 5390 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

Hypothesis

Ref	Expression
funmptd.def	|- ( ph -> F = ( x e. A |-> B ) )

Assertion

Ref	Expression
funmptd	|- ( ph -> Fun F )

Proof of Theorem funmptd

Step	Hyp	Ref	Expression
1		funmpt 5390	. 2 |- Fun ( x e. A |-> B )
2		funmptd.def	. . 3 |- ( ph -> F = ( x e. A |-> B ) )
3	2	funeqd 5374	. 2 |- ( ph -> ( Fun F <-> Fun ( x e. A |-> B ) ) )
4	1, 3	mpbiri 168	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   = wceq 1398   |-> cmpt 4171   Fun wfun 5346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-fun 5354

This theorem is referenced by: (None)