Theorem funmptd 16077

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

Note: one should similarly prove a deduction form of funopab4 5331, then prove funmptd 16077 from it, and then prove funmpt 5332 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 5332	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmptd.def	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
3	2	funeqd 5316	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	1, 3	mpbiri 168	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   = wceq 1375   ↦ cmpt 4124  Fun wfun 5288

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-fun 5296

This theorem is referenced by: (None)