Theorem fnmptf 6016

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypothesis

Ref	Expression
mptfnf.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
fnmptf	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Proof of Theorem fnmptf

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 2952	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		mptfnf.0	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	mptfnf 6015	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	sylib 208	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Syntax hints:   → wi 4   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  offval2f  6909  esumgsum  30107  esumc  30113  bj-mptval  33070  rfovcnvf1od  38298  dssmapf1od  38315  ntrrn  38420  dssmapntrcls  38426