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Theorem pw2f1o 9102
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 2728 . . . 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 9101 . . . . 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 703 . . 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 3475 . . . . 5 𝑦 ∈ V
1211cnvex 7933 . . . 4 𝑦 ∈ V
1312imaex 7922 . . 3 (𝑦 “ {𝐶}) ∈ V
1413a1i 11 . 2 ((𝜑𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (𝑦 “ {𝐶}) ∈ V)
153, 4, 5, 6pw2f1olem 9101 . 2 (𝜑 → ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (𝑦 “ {𝐶}))))
161, 10, 14, 15f1od 7673 1 (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2937  Vcvv 3471  ifcif 4529  𝒫 cpw 4603  {csn 4629  {cpr 4631  cmpt 5231  ccnv 5677  cima 5681  1-1-ontowf1o 6547  (class class class)co 7420  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847
This theorem is referenced by:  pw2eng  9103  indf1o  33643
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