Theorem pwwf 9760

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 9757	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	sspwd 4576	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
3		rankdmr1 9754	. . . . . . 7 ⊢ (rank‘𝐴) ∈ dom 𝑅1
4		r1sucg 9722	. . . . . . 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 3988	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
7		fvex 6871	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
8	7	elpw2 5289	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
9	6, 8	sylibr 234	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
10		r1funlim 9719	. . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
11	10	simpri 485	. . . . . . 7 ⊢ Lim dom 𝑅1
12		limsuc 7825	. . . . . . 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 230	. . . . 5 ⊢ suc (rank‘𝐴) ∈ dom 𝑅1
15		r1sucg 9722	. . . . 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 2839	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
18		r1elwf 9749	. . 3 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
19	17, 18	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
20		r1elssi 9758	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
21		pwexr 7741	. . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ V)
22		pwidg 4583	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
23	21, 22	syl 17	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
24	20, 23	sseldd 3947	. 2 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
25	19, 24	impbii 209	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  dom cdm 5638   “ cima 5641  Oncon0 6332  Lim wlim 6333  suc csuc 6334  Fun wfun 6505  ‘cfv 6511  𝑅₁cr1 9715  rankcrnk 9716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-rank 9718

This theorem is referenced by:  snwf  9762  uniwf  9772  rankpwi  9776  r1pw  9798  r1pwcl  9800  dfac12r  10100  wfgru  10769  xpwf  44954  wfaxsep  44985  wfaxpow  44987