Theorem r1val3 9862

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 9791	. . . . 5 ⊢ 𝑅1 Fn On
2	1	fndmi 6658	. . . 4 ⊢ dom 𝑅1 = On
3	2	eleq2i 2821	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4		r1val1 9810	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
5	3, 4	sylbir 234	. 2 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥))
6		onelon 6394	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
7		r1val2 9861	. . . . 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 4620	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝒫 (𝑅1‘𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
10	9	iuneq2dv 5020	. 2 ⊢ (𝐴 ∈ On → ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
11	5, 10	eqtrd 2768	1 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  𝒫 cpw 4603  ∪ ciun 4996  dom cdm 5678  Oncon0 6369  ‘cfv 6548  𝑅₁cr1 9786  rankcrnk 9787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-reg 9616  ax-inf2 9665

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9788  df-rank 9789

This theorem is referenced by: (None)