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Theorem mapsnend 9021
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 (𝜑𝐴𝑉)
mapsnend.b (𝜑𝐵𝑊)
Assertion
Ref Expression
mapsnend (𝜑 → (𝐴m {𝐵}) ≈ 𝐴)

Proof of Theorem mapsnend
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7429 . 2 (𝜑 → (𝐴m {𝐵}) ∈ V)
2 mapsnend.a . 2 (𝜑𝐴𝑉)
3 fvexd 6894 . . 3 (𝑧 ∈ (𝐴m {𝐵}) → (𝑧𝐵) ∈ V)
43a1i 11 . 2 (𝜑 → (𝑧 ∈ (𝐴m {𝐵}) → (𝑧𝐵) ∈ V))
5 snex 5425 . . 3 {⟨𝐵, 𝑤⟩} ∈ V
652a1i 12 . 2 (𝜑 → (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
7 mapsnend.b . . . . . . 7 (𝜑𝐵𝑊)
82, 7mapsnd 8865 . . . . . 6 (𝜑 → (𝐴m {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
98eqabrd 2876 . . . . 5 (𝜑 → (𝑧 ∈ (𝐴m {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
109anbi1d 630 . . . 4 (𝜑 → ((𝑧 ∈ (𝐴m {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
11 r19.41v 3188 . . . . . 6 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1211bicomi 223 . . . . 5 ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1312a1i 11 . . . 4 (𝜑 → ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
14 df-rex 3071 . . . . 5 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1514a1i 11 . . . 4 (𝜑 → (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
1610, 13, 153bitrd 304 . . 3 (𝜑 → ((𝑧 ∈ (𝐴m {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
17 fveq1 6878 . . . . . . . . . 10 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
18 vex 3478 . . . . . . . . . . 11 𝑦 ∈ V
19 fvsng 7163 . . . . . . . . . . 11 ((𝐵𝑊𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
207, 18, 19sylancl 586 . . . . . . . . . 10 (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2117, 20sylan9eqr 2794 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = 𝑦)
2221eqeq2d 2743 . . . . . . . 8 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
23 equcom 2021 . . . . . . . 8 (𝑤 = 𝑦𝑦 = 𝑤)
2422, 23bitrdi 286 . . . . . . 7 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
2524pm5.32da 579 . . . . . 6 (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
2625anbi2d 629 . . . . 5 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
27 anass 469 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
2827a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
29 ancom 461 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3029a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3126, 28, 303bitr2d 306 . . . 4 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3231exbidv 1924 . . 3 (𝜑 → (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
33 eleq1w 2816 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
34 opeq2 4868 . . . . . . . 8 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
3534sneqd 4635 . . . . . . 7 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
3635eqeq2d 2743 . . . . . 6 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
3733, 36anbi12d 631 . . . . 5 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
3837equsexvw 2008 . . . 4 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
3938a1i 11 . . 3 (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4016, 32, 393bitrd 304 . 2 (𝜑 → ((𝑧 ∈ (𝐴m {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
411, 2, 4, 6, 40en2d 8969 1 (𝜑 → (𝐴m {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3070  Vcvv 3474  {csn 4623  cop 4629   class class class wbr 5142  cfv 6533  (class class class)co 7394  m cmap 8805  cen 8921
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-map 8807  df-en 8925
This theorem is referenced by:  mapsnen  9022  map2xp  9132  mapdom3  9134  ackbij1lem5  10203  pwxpndom2  10644  hashmap  14379  mpct  43735
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