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Theorem mapsnend 8573
 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 7170 . 2 (𝜑 → (𝐴m {𝐵}) ∈ V)
2 mapsnend.a . . 3 (𝜑𝐴𝑉)
32elexd 3461 . 2 (𝜑𝐴 ∈ V)
4 fvexd 6660 . . 3 (𝑧 ∈ (𝐴m {𝐵}) → (𝑧𝐵) ∈ V)
54a1i 11 . 2 (𝜑 → (𝑧 ∈ (𝐴m {𝐵}) → (𝑧𝐵) ∈ V))
6 snex 5297 . . 3 {⟨𝐵, 𝑤⟩} ∈ V
762a1i 12 . 2 (𝜑 → (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
8 mapsnend.b . . . . . . 7 (𝜑𝐵𝑊)
92, 8mapsnd 8435 . . . . . 6 (𝜑 → (𝐴m {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
109abeq2d 2924 . . . . 5 (𝜑 → (𝑧 ∈ (𝐴m {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
1110anbi1d 632 . . . 4 (𝜑 → ((𝑧 ∈ (𝐴m {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
12 r19.41v 3300 . . . . . 6 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1312bicomi 227 . . . . 5 ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1413a1i 11 . . . 4 (𝜑 → ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
15 df-rex 3112 . . . . 5 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1615a1i 11 . . . 4 (𝜑 → (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
1711, 14, 163bitrd 308 . . 3 (𝜑 → ((𝑧 ∈ (𝐴m {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
18 fveq1 6644 . . . . . . . . . 10 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
19 vex 3444 . . . . . . . . . . 11 𝑦 ∈ V
20 fvsng 6919 . . . . . . . . . . 11 ((𝐵𝑊𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
218, 19, 20sylancl 589 . . . . . . . . . 10 (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2218, 21sylan9eqr 2855 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = 𝑦)
2322eqeq2d 2809 . . . . . . . 8 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
24 equcom 2025 . . . . . . . 8 (𝑤 = 𝑦𝑦 = 𝑤)
2523, 24syl6bb 290 . . . . . . 7 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
2625pm5.32da 582 . . . . . 6 (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
2726anbi2d 631 . . . . 5 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
28 anass 472 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
2928a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
30 ancom 464 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3130a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3227, 29, 313bitr2d 310 . . . 4 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3332exbidv 1922 . . 3 (𝜑 → (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
34 eleq1w 2872 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
35 opeq2 4765 . . . . . . . 8 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
3635sneqd 4537 . . . . . . 7 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
3736eqeq2d 2809 . . . . . 6 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
3834, 37anbi12d 633 . . . . 5 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
3938equsexvw 2011 . . . 4 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
4039a1i 11 . . 3 (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4117, 33, 403bitrd 308 . 2 (𝜑 → ((𝑧 ∈ (𝐴m {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
421, 3, 5, 7, 41en2d 8530 1 (𝜑 → (𝐴m {𝐵}) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  {csn 4525  ⟨cop 4531   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135   ↑m cmap 8391   ≈ cen 8491 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 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8393  df-en 8495 This theorem is referenced by:  mapsnen  8574  map2xp  8673  mapdom3  8675  ackbij1lem5  9637  pwxpndom2  10078  hashmap  13794  mpct  41845
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