Theorem pw2f1o 9077

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 2733	. . . 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 9076	. . . . 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 478	. . . 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 703	. . 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 496	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴))
11		vex 3479	. . . . 5 ⊢ 𝑦 ∈ V
12	11	cnvex 7916	. . . 4 ⊢ ◡𝑦 ∈ V
13	12	imaex 7907	. . 3 ⊢ (◡𝑦 “ {𝐶}) ∈ V
14	13	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (◡𝑦 “ {𝐶}) ∈ V)
15	3, 4, 5, 6	pw2f1olem 9076	. 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 7658	1 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  ifcif 4529  𝒫 cpw 4603  {csn 4629  {cpr 4631   ↦ cmpt 5232  ^◡ccnv 5676   “ cima 5680  –1-1-onto→wf1o 6543  (class class class)co 7409   ↑_m cmap 8820

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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822

This theorem is referenced by:  pw2eng  9078  indf1o  33022