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Theorem mapsnend 8188
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 6825 . 2 (𝜑 → (𝐴𝑚 {𝐵}) ∈ V)
2 mapsnend.a . . 3 (𝜑𝐴𝑉)
32elexd 3366 . 2 (𝜑𝐴 ∈ V)
4 fvexd 6344 . . 3 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
54a1i 11 . 2 (𝜑 → (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V))
6 snex 5036 . . . 4 {⟨𝐵, 𝑤⟩} ∈ V
76a1i 11 . . 3 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
87a1i 11 . 2 (𝜑 → (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
9 mapsnend.b . . . . . . 7 (𝜑𝐵𝑊)
102, 9mapsnd 8051 . . . . . 6 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
1110abeq2d 2883 . . . . 5 (𝜑 → (𝑧 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
1211anbi1d 615 . . . 4 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
13 r19.41v 3237 . . . . . 6 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1413bicomi 214 . . . . 5 ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1514a1i 11 . . . 4 (𝜑 → ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
16 df-rex 3067 . . . . 5 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1716a1i 11 . . . 4 (𝜑 → (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
1812, 15, 173bitrd 294 . . 3 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
19 fveq1 6331 . . . . . . . . . . . 12 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
2019adantl 467 . . . . . . . . . . 11 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21 vex 3354 . . . . . . . . . . . . . 14 𝑦 ∈ V
2221a1i 11 . . . . . . . . . . . . 13 (𝜑𝑦 ∈ V)
23 fvsng 6591 . . . . . . . . . . . . 13 ((𝐵𝑊𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
249, 22, 23syl2anc 573 . . . . . . . . . . . 12 (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2524adantr 466 . . . . . . . . . . 11 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2620, 25eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = 𝑦)
2726eqeq2d 2781 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
28 equcom 2103 . . . . . . . . . 10 (𝑤 = 𝑦𝑦 = 𝑤)
2928a1i 11 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = 𝑦𝑦 = 𝑤))
3027, 29bitrd 268 . . . . . . . 8 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
3130ex 397 . . . . . . 7 (𝜑 → (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤)))
3231pm5.32d 566 . . . . . 6 (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
3332anbi2d 614 . . . . 5 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
34 anass 459 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
3534a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
36 ancom 452 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3736a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3833, 35, 373bitr2d 296 . . . 4 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3938exbidv 2002 . . 3 (𝜑 → (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
40 vex 3354 . . . . 5 𝑤 ∈ V
41 eleq1w 2833 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
42 opeq2 4540 . . . . . . . 8 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
4342sneqd 4328 . . . . . . 7 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
4443eqeq2d 2781 . . . . . 6 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
4541, 44anbi12d 616 . . . . 5 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4640, 45ceqsexv 3394 . . . 4 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
4746a1i 11 . . 3 (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4818, 39, 473bitrd 294 . 2 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
491, 3, 5, 8, 48en2d 8145 1 (𝜑 → (𝐴𝑚 {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wrex 3062  Vcvv 3351  {csn 4316  cop 4322   class class class wbr 4786  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  cen 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-en 8110
This theorem is referenced by:  mapsnen  8189  map2xp  8286  mapdom3  8288  ackbij1lem5  9248  pwxpndom2  9689  hashmap  13424  mpct  39911
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