Theorem mapsnend 8826

Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
mapsnend.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mapsnend.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
mapsnend	⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴)

Proof of Theorem mapsnend

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7310	. 2 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ∈ V)
2		mapsnend.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fvexd 6789	. . 3 ⊢ (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V)
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V))
5		snex 5354	. . 3 ⊢ {⟨𝐵, 𝑤⟩} ∈ V
6	5	2a1i 12	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
7		mapsnend.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8	2, 7	mapsnd 8674	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
9	8	abeq2d 2874	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
10	9	anbi1d 630	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
11		r19.41v 3276	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
12	11	bicomi 223	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
13	12	a1i 11	. . . 4 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
14		df-rex 3070	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
15	14	a1i 11	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))))
16	10, 13, 15	3bitrd 305	. . 3 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))))
17		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
18		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
19		fvsng 7052	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
20	7, 18, 19	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
21	17, 20	sylan9eqr 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧‘𝐵) = 𝑦)
22	21	eqeq2d 2749	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
23		equcom 2021	. . . . . . . 8 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
24	22, 23	bitrdi 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))
25	24	pm5.32da 579	. . . . . 6 ⊢ (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
26	25	anbi2d 629	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
27		anass 469	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
28	27	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
29		ancom 461	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
31	26, 28, 30	3bitr2d 307	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
32	31	exbidv 1924	. . 3 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
33		eleq1w 2821	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
34		opeq2 4805	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
35	34	sneqd 4573	. . . . . . 7 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
36	35	eqeq2d 2749	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
37	33, 36	anbi12d 631	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
38	37	equsexvw 2008	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
39	38	a1i 11	. . 3 ⊢ (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
40	16, 32, 39	3bitrd 305	. 2 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
41	1, 2, 4, 6, 40	en2d 8776	1 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432  {csn 4561  ⟨cop 4567   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-en 8734

This theorem is referenced by:  mapsnen  8827  map2xp  8934  mapdom3  8936  ackbij1lem5  9980  pwxpndom2  10421  hashmap  14150  mpct  42741