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Theorem pw2f1o 8106
Description: The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
Hypotheses
Ref Expression
pw2f1o.1 (𝜑𝐴𝑉)
pw2f1o.2 (𝜑𝐵𝑊)
pw2f1o.3 (𝜑𝐶𝑊)
pw2f1o.4 (𝜑𝐵𝐶)
pw2f1o.5 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)))
Assertion
Ref Expression
pw2f1o (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑𝑚 𝐴))
Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝐵,𝑧   𝑥,𝐶,𝑧   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑧)   𝐹(𝑥,𝑧)   𝑉(𝑥,𝑧)   𝑊(𝑥,𝑧)

Proof of Theorem pw2f1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pw2f1o.5 . 2 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)))
2 eqid 2651 . . . 4 (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))
3 pw2f1o.1 . . . . . 6 (𝜑𝐴𝑉)
4 pw2f1o.2 . . . . . 6 (𝜑𝐵𝑊)
5 pw2f1o.3 . . . . . 6 (𝜑𝐶𝑊)
6 pw2f1o.4 . . . . . 6 (𝜑𝐵𝐶)
73, 4, 5, 6pw2f1olem 8105 . . . . 5 (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))) ↔ ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑥 = ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) “ {𝐶}))))
87biimpa 500 . . . 4 ((𝜑 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)))) → ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑥 = ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) “ {𝐶})))
92, 8mpanr2 720 . . 3 ((𝜑𝑥 ∈ 𝒫 𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑥 = ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) “ {𝐶})))
109simpld 474 . 2 ((𝜑𝑥 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑𝑚 𝐴))
11 vex 3234 . . . . 5 𝑦 ∈ V
1211cnvex 7155 . . . 4 𝑦 ∈ V
1312imaex 7146 . . 3 (𝑦 “ {𝐶}) ∈ V
1413a1i 11 . 2 ((𝜑𝑦 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴)) → (𝑦 “ {𝐶}) ∈ V)
153, 4, 5, 6pw2f1olem 8105 . 2 (𝜑 → ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑥 = (𝑦 “ {𝐶}))))
161, 10, 14, 15f1od 6927 1 (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  ifcif 4119  𝒫 cpw 4191  {csn 4210  {cpr 4212  cmpt 4762  ccnv 5142  cima 5146  1-1-ontowf1o 5925  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901
This theorem is referenced by:  pw2eng  8107  indf1o  30214
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