Theorem r1val3 9876

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 9805	. . . . 5 ⊢ 𝑅1 Fn On
2	1	fndmi 6673	. . . 4 ⊢ dom 𝑅1 = On
3	2	eleq2i 2831	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4		r1val1 9824	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
5	3, 4	sylbir 235	. 2 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
6		onelon 6411	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
7		r1val2 9875	. . . . 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 4622	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
10	9	iuneq2dv 5021	. 2 ⊢ (𝐴 ∈ On → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
11	5, 10	eqtrd 2775	1 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  𝒫 cpw 4605  ∪ ciun 4996  dom cdm 5689  Oncon0 6386  ‘cfv 6563  𝑅₁cr1 9800  rankcrnk 9801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803

This theorem is referenced by: (None)