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Theorem pw2f1o 8614
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 2819 . . . 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 8613 . . . . 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 479 . . . 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 702 . . 3 ((𝜑𝑥 ∈ 𝒫 𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = ((𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) “ {𝐶})))
109simpld 497 . 2 ((𝜑𝑥 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴))
11 vex 3496 . . . . 5 𝑦 ∈ V
1211cnvex 7622 . . . 4 𝑦 ∈ V
1312imaex 7613 . . 3 (𝑦 “ {𝐶}) ∈ V
1413a1i 11 . 2 ((𝜑𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (𝑦 “ {𝐶}) ∈ V)
153, 4, 5, 6pw2f1olem 8613 . 2 (𝜑 → ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (𝑦 “ {𝐶}))))
161, 10, 14, 15f1od 7389 1 (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  wne 3014  Vcvv 3493  ifcif 4465  𝒫 cpw 4537  {csn 4559  {cpr 4561  cmpt 5137  ccnv 5547  cima 5551  1-1-ontowf1o 6347  (class class class)co 7148  m cmap 8398
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400
This theorem is referenced by:  pw2eng  8615  indf1o  31276
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