Theorem r1val3 9305

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
r1val3	⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem r1val3

Step	Hyp	Ref	Expression
1		r1fnon 9234	. . . . 5 ⊢ 𝑅1 Fn On
2	1	fndmi 6441	. . . 4 ⊢ dom 𝑅1 = On
3	2	eleq2i 2843	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4		r1val1 9253	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
5	3, 4	sylbir 238	. 2 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
6		onelon 6198	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
7		r1val2 9304	. . . . 5 ⊢ (𝑥 ∈ On → (𝑅1‘𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (𝑅1‘𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
9	8	pweqd 4516	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
10	9	iuneq2dv 4910	. 2 ⊢ (𝐴 ∈ On → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
11	5, 10	eqtrd 2793	1 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  𝒫 cpw 4497  ∪ ciun 4886  dom cdm 5527  Oncon0 6173  ‘cfv 6339  𝑅₁cr1 9229  rankcrnk 9230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-reg 9094  ax-inf2 9142

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7585  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-r1 9231  df-rank 9232

This theorem is referenced by: (None)