Theorem pw2f1odc 6992

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)
pw2f1odc.4	⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞)
pw2f1o.5	⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)))

Assertion

Ref	Expression
pw2f1odc	⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑𝑚 𝐴))

Distinct variable groups:   𝐴,𝑝,𝑞,𝑥   𝑧,𝐴,𝑥   𝑥,𝐵,𝑧   𝑥,𝐶,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑧,𝑞,𝑝)   𝐵(𝑞,𝑝)   𝐶(𝑞,𝑝)   𝐹(𝑥,𝑧,𝑞,𝑝)   𝑉(𝑥,𝑧,𝑞,𝑝)   𝑊(𝑥,𝑧,𝑞,𝑝)

Proof of Theorem pw2f1odc

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 2229	. . . 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		pw2f1odc.4	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞)
8	3, 4, 5, 6, 7	pw2f1odclem 6991	. . . . 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}))))
9	8	biimpa 296	. . . 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})))
10	2, 9	mpanr2 438	. . 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})))
11	10	simpld 112	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑𝑚 𝐴))
12		vex 2802	. . . . 5 ⊢ 𝑦 ∈ V
13	12	cnvex 5266	. . . 4 ⊢ ◡𝑦 ∈ V
14	13	imaex 5082	. . 3 ⊢ (◡𝑦 “ {𝐶}) ∈ V
15	14	a1i 9	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴)) → (◡𝑦 “ {𝐶}) ∈ V)
16	3, 4, 5, 6, 7	pw2f1odclem 6991	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑥 = (◡𝑦 “ {𝐶}))))
17	1, 11, 15, 16	f1od 6207	1 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑𝑚 𝐴))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  Vcvv 2799  ifcif 3602  𝒫 cpw 3649  {csn 3666  {cpr 3667   ↦ cmpt 4144  ^◡ccnv 4717   “ cima 4721  –1-1-onto→wf1o 5316  (class class class)co 6000   ↑_𝑚 cmap 6793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-map 6795

This theorem is referenced by:  exmidpw2en  7070