Theorem funmptd 16191

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

Note: one should similarly prove a deduction form of funopab4 5355, then prove funmptd 16191 from it, and then prove funmpt 5356 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

Hypothesis

Ref	Expression
funmptd.def	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))

Assertion

Ref	Expression
funmptd	⊢ (𝜑 → Fun 𝐹)

Proof of Theorem funmptd

Step	Hyp	Ref	Expression
1		funmpt 5356	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmptd.def	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
3	2	funeqd 5340	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	1, 3	mpbiri 168	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   = wceq 1395   ↦ cmpt 4145  Fun wfun 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-fun 5320

This theorem is referenced by: (None)