Theorem mptfnf 6673

Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypothesis

Ref	Expression
mptfnf.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
mptfnf	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Proof of Theorem mptfnf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eueq 3691	. . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
2	1	ralbii 3082	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵)
3		r19.26 3098	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵))
4		df-eu 2568	. . . 4 ⊢ (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
5	4	ralbii 3082	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6		df-mpt 5202	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7	6	fneq1i 6635	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} Fn 𝐴)
8		df-fn 6534	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴))
9	7, 8	bitri 275	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴))
10		moanimv 2618	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
11	10	albii 1819	. . . . . 6 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
12		funopab 6571	. . . . . 6 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
13		df-ral 3052	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
14	11, 12, 13	3bitr4ri 304	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)})
15		eqcom 2742	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴 ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16		dmopab 5895	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
17		19.42v 1953	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
18	17	abbii 2802	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
19	16, 18	eqtri 2758	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
20	19	eqeq1i 2740	. . . . . 6 ⊢ (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21		pm4.71 557	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
22	21	albii 1819	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23		df-ral 3052	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵))
24		mptfnf.0	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
25	24	eqabf 2928	. . . . . . 7 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
26	22, 23, 25	3bitr4i 303	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
27	15, 20, 26	3bitr4ri 304	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴)
28	14, 27	anbi12i 628	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴))
29		ancom 460	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵))
30	9, 28, 29	3bitr2i 299	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵))
31	3, 5, 30	3bitr4ri 304	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵)
32	2, 31	bitr4i 278	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃*wmo 2537  ∃!weu 2567  {cab 2713  Ⅎwnfc 2883  ∀wral 3051  Vcvv 3459  {copab 5181   ↦ cmpt 5201  dom cdm 5654  Fun wfun 6525   Fn wfn 6526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-fun 6533  df-fn 6534

This theorem is referenced by:  fnmptf  6674  mptfnd  45266  fnmptif  45289