Theorem funmptd 15365

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

Note: one should similarly prove a deduction form of funopab4 5292, then prove funmptd 15365 from it, and then prove funmpt 5293 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 5293	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmptd.def	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
3	2	funeqd 5277	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	1, 3	mpbiri 168	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   = wceq 1364   ↦ cmpt 4091  Fun wfun 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-fun 5257

This theorem is referenced by: (None)