Theorem r1val3 9844

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 9773	. . . . 5 ⊢ 𝑅1 Fn On
2	1	fndmi 6638	. . . 4 ⊢ dom 𝑅1 = On
3	2	eleq2i 2825	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4		r1val1 9792	. . 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 6374	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
7		r1val2 9843	. . . . 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 4590	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
10	9	iuneq2dv 4989	. 2 ⊢ (𝐴 ∈ On → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
11	5, 10	eqtrd 2769	1 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2712  𝒫 cpw 4573  ∪ ciun 4964  dom cdm 5651  Oncon0 6349  ‘cfv 6527  𝑅₁cr1 9768  rankcrnk 9769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-reg 9598  ax-inf2 9647

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-int 4920  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-om 7856  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-r1 9770  df-rank 9771

This theorem is referenced by: (None)