Theorem pw2f1o 9071

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 2763	. . . 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 9070	. . . . 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 481	. . . 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 716	. . 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 499	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴))
11		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
12	11	cnvex 7923	. . . 4 ⊢ ◡𝑦 ∈ V
13	12	imaex 7912	. . 3 ⊢ (◡𝑦 “ {𝐶}) ∈ V
14	13	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (◡𝑦 “ {𝐶}) ∈ V)
15	3, 4, 5, 6	pw2f1olem 9070	. 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 7664	1 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ifcif 4488  𝒫 cpw 4563  {csn 4590  {cpr 4592   ↦ cmpt 5193  ^◡ccnv 5662   “ cima 5666  –1-1-onto→wf1o 6537  (class class class)co 7412   ↑_m cmap 8825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827

This theorem is referenced by:  pw2eng  9072  indf1o  33162