Theorem pwwf 9792

Description: A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
pwwf	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem pwwf

Step	Hyp	Ref	Expression
1		r1rankidb 9789	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	sspwd 4573	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
3		rankdmr1 9786	. . . . . . 7 ⊢ (rank‘𝐴) ∈ dom 𝑅1
4		r1sucg 9754	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
6	2, 5	sseqtrrdi 3975	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
7		fvex 6895	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
8	7	elpw2 5303	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
9	6, 8	sylibr 237	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
10		r1funlim 9751	. . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
11	10	simpri 491	. . . . . . 7 ⊢ Lim dom 𝑅1
12		limsuc 7848	. . . . . . 7 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)
14	3, 13	mpbi 233	. . . . 5 ⊢ suc (rank‘𝐴) ∈ dom 𝑅1
15		r1sucg 9754	. . . . 5 ⊢ (suc (rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
16	14, 15	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
17	9, 16	eleqtrrdi 2873	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
18		r1elwf 9781	. . 3 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
19	17, 18	syl 18	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
20		r1elssi 9790	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
21		pwexr 7767	. . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ V)
22		pwidg 4580	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
23	21, 22	syl 18	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
24	20, 23	sseldd 3935	. 2 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
25	19, 24	impbii 212	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  dom cdm 5659   “ cima 5662  Oncon0 6361  Lim wlim 6362  suc csuc 6363  Fun wfun 6531  ‘cfv 6537  𝑅₁cr1 9747  rankcrnk 9748

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749  df-rank 9750

This theorem is used by:  snwf  9794  uniwf  9804  rankpwi  9808  r1pw  9830  r1pwcl  9832  dfac12r  10152  wfgru  10828  xpwf  45774  wfaxsep  45805  wfaxpow  45807