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Theorem pw2f1o 9118
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 2736 . . . 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 9117 . . . . 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 476 . . . 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 704 . . 3 ((𝜑𝑥 ∈ 𝒫 𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) “ {𝐶})))
109simpld 494 . 2 ((𝜑𝑥 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴))
11 vex 3483 . . . . 5 𝑦 ∈ V
1211cnvex 7948 . . . 4 𝑦 ∈ V
1312imaex 7937 . . 3 (𝑦 “ {𝐶}) ∈ V
1413a1i 11 . 2 ((𝜑𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (𝑦 “ {𝐶}) ∈ V)
153, 4, 5, 6pw2f1olem 9117 . 2 (𝜑 → ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (𝑦 “ {𝐶}))))
161, 10, 14, 15f1od 7686 1 (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wne 2939  Vcvv 3479  ifcif 4524  𝒫 cpw 4599  {csn 4625  {cpr 4627  cmpt 5224  ccnv 5683  cima 5687  1-1-ontowf1o 6559  (class class class)co 7432  m cmap 8867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869
This theorem is referenced by:  pw2eng  9119  indf1o  32850
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