Theorem r1val3 3523

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113.

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem r1val3

Step	Hyp	Ref	Expression
1		r1val1 3502	. 2 ⊢ (A ∈ On → (R1 ‘A) = ∪x ∈ A ℘(R1 ‘x))
2		onelon 2223	. . . . . 6 ⊢ ((A ∈ On ∧ x ∈ A) → x ∈ On)
3		r1val2 3522	. . . . . 6 ⊢ (x ∈ On → (R1 ‘x) = {y∣(rank ‘y) ∈ x})
4		pweq 1800	. . . . . 6 ⊢ ((R1 ‘x) = {y∣(rank ‘y) ∈ x} → ℘(R1 ‘x) = ℘{y∣(rank ‘y) ∈ x})
5	2, 3, 4	3syl 21	. . . . 5 ⊢ ((A ∈ On ∧ x ∈ A) → ℘(R1 ‘x) = ℘{y∣(rank ‘y) ∈ x})
6	5	exp 291	. . . 4 ⊢ (A ∈ On → (x ∈ A → ℘(R1 ‘x) = ℘{y∣(rank ‘y) ∈ x}))
7	6	r19.21aiv 1259	. . 3 ⊢ (A ∈ On → ∀x ∈ A ℘(R1 ‘x) = ℘{y∣(rank ‘y) ∈ x})
8		iuneq2 2006	. . 3 ⊢ (∀x ∈ A ℘(R1 ‘x) = ℘{y∣(rank ‘y) ∈ x} → ∪x ∈ A ℘(R1 ‘x) = ∪x ∈ A ℘{y∣(rank ‘y) ∈ x})
9	7, 8	syl 12	. 2 ⊢ (A ∈ On → ∪x ∈ A ℘(R1 ‘x) = ∪x ∈ A ℘{y∣(rank ‘y) ∈ x})
10	1, 9	eqtrd 1128	1 ⊢ (A ∈ On → (R1 ‘A) = ∪x ∈ A ℘{y∣(rank ‘y) ∈ x})

Syntax hints:   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ℘cpw 1798  ∪ciun 1994  Oncon0 2199   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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