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Theorem mapsnen 7898
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.)
Hypotheses
Ref Expression
mapsnen.1 𝐴 ∈ V
mapsnen.2 𝐵 ∈ V
Assertion
Ref Expression
mapsnen (𝐴𝑚 {𝐵}) ≈ 𝐴

Proof of Theorem mapsnen
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6555 . 2 (𝐴𝑚 {𝐵}) ∈ V
2 mapsnen.1 . 2 𝐴 ∈ V
3 fvex 6098 . . 3 (𝑧𝐵) ∈ V
43a1i 11 . 2 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
5 snex 4830 . . 3 {⟨𝐵, 𝑤⟩} ∈ V
65a1i 11 . 2 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
7 mapsnen.2 . . . . . . 7 𝐵 ∈ V
82, 7mapsn 7763 . . . . . 6 (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
98abeq2i 2722 . . . . 5 (𝑧 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩})
109anbi1i 727 . . . 4 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
11 r19.41v 3070 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
12 df-rex 2902 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1310, 11, 123bitr2i 287 . . 3 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
14 fveq1 6087 . . . . . . . . . 10 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
15 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
167, 15fvsn 6329 . . . . . . . . . 10 ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
1714, 16syl6eq 2660 . . . . . . . . 9 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = 𝑦)
1817eqeq2d 2620 . . . . . . . 8 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
19 equcom 1932 . . . . . . . 8 (𝑤 = 𝑦𝑦 = 𝑤)
2018, 19syl6bb 275 . . . . . . 7 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
2120pm5.32i 667 . . . . . 6 ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))
2221anbi2i 726 . . . . 5 ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
23 anass 679 . . . . 5 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
24 ancom 465 . . . . 5 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
2522, 23, 243bitr2i 287 . . . 4 ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
2625exbii 1764 . . 3 (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
27 vex 3176 . . . 4 𝑤 ∈ V
28 eleq1 2676 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
29 opeq2 4336 . . . . . . 7 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
3029sneqd 4137 . . . . . 6 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
3130eqeq2d 2620 . . . . 5 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
3228, 31anbi12d 743 . . . 4 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
3327, 32ceqsexv 3215 . . 3 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
3413, 26, 333bitri 285 . 2 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
351, 2, 4, 6, 34en2i 7857 1 (𝐴𝑚 {𝐵}) ≈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wex 1695  wcel 1977  wrex 2897  Vcvv 3173  {csn 4125  cop 4131   class class class wbr 4578  cfv 5790  (class class class)co 6527  𝑚 cmap 7722  cen 7816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724  df-en 7820
This theorem is referenced by:  map2xp  7993  mapdom3  7995  ackbij1lem5  8907  pwxpndom2  9344  hashmap  13037
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