Theorem pw2f1o 9014

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	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pw2f1o.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
pw2f1o.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)
pw2f1o.4	⊢ (𝜑 → 𝐵 ≠ 𝐶)
pw2f1o.5	⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)))

Assertion

Ref	Expression
pw2f1o	⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-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.

Step	Hyp	Ref	Expression
1		pw2f1o.5	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)))
2		eqid 2737	. . . 4 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))
3		pw2f1o.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		pw2f1o.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		pw2f1o.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
6		pw2f1o.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
7	3, 4, 5, 6	pw2f1olem 9013	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ↔ ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡(𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) “ {𝐶}))))
8	7	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)))) → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡(𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) “ {𝐶})))
9	2, 8	mpanr2 705	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡(𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) “ {𝐶})))
10	9	simpld 494	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴))
11		vex 3445	. . . . 5 ⊢ 𝑦 ∈ V
12	11	cnvex 7869	. . . 4 ⊢ ◡𝑦 ∈ V
13	12	imaex 7858	. . 3 ⊢ (◡𝑦 “ {𝐶}) ∈ V
14	13	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (◡𝑦 “ {𝐶}) ∈ V)
15	3, 4, 5, 6	pw2f1olem 9013	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡𝑦 “ {𝐶}))))
16	1, 10, 14, 15	f1od 7612	1 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3441  ifcif 4480  𝒫 cpw 4555  {csn 4581  {cpr 4583   ↦ cmpt 5180  ^◡ccnv 5624   “ cima 5628  –1-1-onto→wf1o 6492  (class class class)co 7360   ↑_m cmap 8767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769

This theorem is referenced by:  pw2eng  9015  indf1o  32927