Theorem pwwf 8708

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 8705	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		sspwb 4947	. . . . . . 7 ⊢ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
3	1, 2	sylib 208	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
4		rankdmr1 8702	. . . . . . 7 ⊢ (rank‘𝐴) ∈ dom 𝑅1
5		r1sucg 8670	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
7	3, 6	syl6sseqr 3685	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
8		fvex 6239	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
9	8	elpw2 4858	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
10	7, 9	sylibr 224	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
11		r1funlim 8667	. . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
12	11	simpri 477	. . . . . . 7 ⊢ Lim dom 𝑅1
13		limsuc 7091	. . . . . . 7 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1))
14	12, 13	ax-mp 5	. . . . . 6 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)
15	4, 14	mpbi 220	. . . . 5 ⊢ suc (rank‘𝐴) ∈ dom 𝑅1
16		r1sucg 8670	. . . . 5 ⊢ (suc (rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
17	15, 16	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
18	10, 17	syl6eleqr 2741	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
19		r1elwf 8697	. . 3 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
20	18, 19	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
21		r1elssi 8706	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
22		pwexr 7016	. . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ V)
23		pwidg 4206	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
24	22, 23	syl 17	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
25	21, 24	sseldd 3637	. 2 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
26	20, 25	impbii 199	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 196   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468  dom cdm 5143   “ cima 5146  Oncon0 5761  Lim wlim 5762  suc csuc 5763  Fun wfun 5920  ‘cfv 5926  𝑅₁cr1 8663  rankcrnk 8664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666

This theorem is referenced by:  snwf  8710  uniwf  8720  rankpwi  8724  r1pw  8746  r1pwcl  8748  dfac12r  9006  wfgru  9676