Theorem funmptd 13338

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

Note: one should similarly prove a deduction form of funopab4 5204, then prove funmptd 13338 from it, and then prove funmpt 5205 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 5205	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmptd.def	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
3	2	funeqd 5189	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	1, 3	mpbiri 167	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   = wceq 1335   ↦ cmpt 4025  Fun wfun 5161

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-fun 5169

This theorem is referenced by: (None)