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Theorem pw2f1o 9028
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 2737 . . . 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 9027 . . . . 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 478 . . . 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 496 . 2 ((𝜑𝑥 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴))
11 vex 3452 . . . . 5 𝑦 ∈ V
1211cnvex 7867 . . . 4 𝑦 ∈ V
1312imaex 7858 . . 3 (𝑦 “ {𝐶}) ∈ V
1413a1i 11 . 2 ((𝜑𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (𝑦 “ {𝐶}) ∈ V)
153, 4, 5, 6pw2f1olem 9027 . 2 (𝜑 → ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (𝑦 “ {𝐶}))))
161, 10, 14, 15f1od 7610 1 (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2944  Vcvv 3448  ifcif 4491  𝒫 cpw 4565  {csn 4591  {cpr 4593  cmpt 5193  ccnv 5637  cima 5641  1-1-ontowf1o 6500  (class class class)co 7362  m cmap 8772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774
This theorem is referenced by:  pw2eng  9029  indf1o  32663
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