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Theorem mapsnen 8200
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 6841 . 2 (𝐴𝑚 {𝐵}) ∈ V
2 mapsnen.1 . 2 𝐴 ∈ V
3 fvexd 6364 . 2 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
4 snex 5057 . . 3 {⟨𝐵, 𝑤⟩} ∈ V
54a1i 11 . 2 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
6 mapsnen.2 . . . . . . 7 𝐵 ∈ V
72, 6mapsn 8065 . . . . . 6 (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
87abeq2i 2873 . . . . 5 (𝑧 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩})
98anbi1i 733 . . . 4 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
10 r19.41v 3227 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
11 df-rex 3056 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
129, 10, 113bitr2i 288 . . 3 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
13 fveq1 6351 . . . . . . . . . 10 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
14 vex 3343 . . . . . . . . . . 11 𝑦 ∈ V
156, 14fvsn 6610 . . . . . . . . . 10 ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
1613, 15syl6eq 2810 . . . . . . . . 9 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = 𝑦)
1716eqeq2d 2770 . . . . . . . 8 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
18 equcom 2100 . . . . . . . 8 (𝑤 = 𝑦𝑦 = 𝑤)
1917, 18syl6bb 276 . . . . . . 7 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
2019pm5.32i 672 . . . . . 6 ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))
2120anbi2i 732 . . . . 5 ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
22 anass 684 . . . . 5 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
23 ancom 465 . . . . 5 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
2421, 22, 233bitr2i 288 . . . 4 ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
2524exbii 1923 . . 3 (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
26 vex 3343 . . . 4 𝑤 ∈ V
27 eleq1w 2822 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
28 opeq2 4554 . . . . . . 7 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
2928sneqd 4333 . . . . . 6 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
3029eqeq2d 2770 . . . . 5 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
3127, 30anbi12d 749 . . . 4 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
3226, 31ceqsexv 3382 . . 3 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
3312, 25, 323bitri 286 . 2 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
341, 2, 3, 5, 33en2i 8159 1 (𝐴𝑚 {𝐵}) ≈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  wrex 3051  Vcvv 3340  {csn 4321  cop 4327   class class class wbr 4804  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  cen 8118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-en 8122
This theorem is referenced by:  map2xp  8295  mapdom3  8297  ackbij1lem5  9238  pwxpndom2  9679  hashmap  13414
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