Theorem mptfnf 6674

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 3673	. . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
2	1	ralbii 3113	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵)
3		r19.26 3127	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵))
4		df-eu 2599	. . . 4 ⊢ (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
5	4	ralbii 3113	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6		df-mpt 5195	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7	6	fneq1i 6636	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} Fn 𝐴)
8		df-fn 6543	. . . . 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 278	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴))
10		moanimv 2649	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
11	10	albii 1852	. . . . . 6 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
12		funopab 6575	. . . . . 6 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
13		df-ral 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
14	11, 12, 13	3bitr4ri 307	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)})
15		eqcom 2772	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴 ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16		dmopab 5907	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
17		19.42v 1986	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
18	17	abbii 2832	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
19	16, 18	eqtri 2788	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
20	19	eqeq1i 2770	. . . . . 6 ⊢ (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21		pm4.71 567	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
22	21	albii 1852	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23		df-ral 3082	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵))
24		mptfnf.0	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
25	24	eqabf 2956	. . . . . . 7 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
26	22, 23, 25	3bitr4i 306	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
27	15, 20, 26	3bitr4ri 307	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴)
28	14, 27	anbi12i 640	. . . 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 466	. . . 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 302	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵))
31	3, 5, 30	3bitr4ri 307	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵)
32	2, 31	bitr4i 281	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2567  ∃!weu 2598  {cab 2743  Ⅎwnfc 2912  ∀wral 3081  Vcvv 3457  {copab 5175   ↦ cmpt 5194  dom cdm 5663  Fun wfun 6534   Fn wfn 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-fun 6542  df-fn 6543

This theorem is used by:  fnmptf  6675  mptfnd  45990  fnmptif  46013  sqrtnnaa  47637  sqrtnzqaa  47638  sinnpoly  47661