Theorem mptfng 5483

Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)

Hypothesis

Ref	Expression
mptfng.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
mptfng	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mptfng

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eueq 2987	. . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
2	1	ralbii 2548	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵)
3		mptfng.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		df-mpt 4172	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5	3, 4	eqtri 2253	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	5	fnopabg 5481	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ 𝐹 Fn 𝐴)
7	2, 6	bitri 184	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃!weu 2080   ∈ wcel 2203  ∀wral 2520  Vcvv 2812  {copab 4169   ↦ cmpt 4170   Fn wfn 5346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-fun 5353  df-fn 5354

This theorem is referenced by:  fnmpt  5484  fnmpti  5486  mpteqb  5767  cc3  7578