Theorem funmptd 13685

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

Note: one should similarly prove a deduction form of funopab4 5225, then prove funmptd 13685 from it, and then prove funmpt 5226 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 5226	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmptd.def	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
3	2	funeqd 5210	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)))
4	1, 3	mpbiri 167	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   = wceq 1343   ↦ cmpt 4043  Fun wfun 5182

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-fun 5190

This theorem is referenced by: (None)