Theorem mapsnend 9046

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 7449	. 2 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ∈ V)
2		mapsnend.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fvexd 6894	. . 3 ⊢ (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V)
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V))
5		snex 5404	. . 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 8896	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
9	8	eqabrd 2901	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
10	9	anbi1d 643	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
11		r19.41v 3192	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
12	11	bicomi 227	. . . . 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 3087	. . . . 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 308	. . 3 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))))
17		fveq1 6878	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
18		vex 3454	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
19		fvsng 7179	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
20	7, 18, 19	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
21	17, 20	sylan9eqr 2817	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧‘𝐵) = 𝑦)
22	21	eqeq2d 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
23		equcom 2051	. . . . . . . 8 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
24	22, 23	bitrdi 290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))
25	24	pm5.32da 590	. . . . . 6 ⊢ (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
26	25	anbi2d 642	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
27		anass 474	. . . . . 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 466	. . . . . 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 310	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
32	31	exbidv 1954	. . 3 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
33		eleq1w 2843	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
34		opeq2 4834	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
35	34	sneqd 4596	. . . . . . 7 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
36	35	eqeq2d 2771	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
37	33, 36	anbi12d 644	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
38	37	equsexvw 2038	. . . 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 308	. 2 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
41	1, 2, 4, 6, 40	en2d 8997	1 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450  {csn 4584  ⟨cop 4590   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414   ↑_m cmap 8829   ≈ cen 8952

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8831  df-en 8956

This theorem is used by:  mapsnen  9047  map2xp  9148  mapdom3  9150  ackbij1lem5  10228  pwxpndom2  10677  hashmap  14503  mpct  46035