Theorem mptfnf 6471

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 3624	. . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
2	1	ralbii 3097	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵)
3		r19.26 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑦 = 𝐵))
4		df-eu 2588	. . . 4 ⊢ (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
5	4	ralbii 3097	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6		df-mpt 5117	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7	6	fneq1i 6436	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} Fn 𝐴)
8		df-fn 6343	. . . . 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 2640	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
11	10	albii 1821	. . . . . 6 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑦 = 𝐵))
12		funopab 6375	. . . . . 6 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
13		df-ral 3075	. . . . . 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 2765	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴 ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16		dmopab 5761	. . . . . . . 8 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
17		19.42v 1954	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
18	17	abbii 2823	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
19	16, 18	eqtri 2781	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
20	19	eqeq1i 2763	. . . . . 6 ⊢ (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21		pm4.71 561	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
22	21	albii 1821	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23		df-ral 3075	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦 𝑦 = 𝐵))
24		mptfnf.0	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
25	24	abeq2f 2949	. . . . . . 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 629	. . . 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 464	. . . 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2555  ∃!weu 2587  {cab 2735  Ⅎwnfc 2899  ∀wral 3070  Vcvv 3409  {copab 5098   ↦ cmpt 5116  dom cdm 5528  Fun wfun 6334   Fn wfn 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-fun 6342  df-fn 6343

This theorem is referenced by:  fnmptf  6472  mptfnd  42281