Theorem funmptd 16745

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

Note: one should similarly prove a deduction form of funopab4 5409, then prove funmptd 16745 from it, and then prove funmpt 5410 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 5410	. 2 |- Fun ( x e. A |-> B )
2		funmptd.def	. . 3 |- ( ph -> F = ( x e. A |-> B ) )
3	2	funeqd 5394	. 2 |- ( ph -> ( Fun F <-> Fun ( x e. A |-> B ) ) )
4	1, 3	mpbiri 168	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   = wceq 1402   |-> cmpt 4187   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-fun 5374

This theorem is referenced by: (None)