Theorem rdglimt 2986

Description: The value of the recursive definition generator at a limit ordinal.

Assertion

Ref	Expression
rdglimt	⊢ ((B ∈ C ∧ Lim B) → (rec(F, A) ‘B) = ∪(rec(F, A) “ B))

Proof of Theorem rdglimt

Step	Hyp	Ref	Expression
1		limeq 2211	. . . . . 6 ⊢ (B = if(B ∈ On, B, ∅) → (Lim B ↔ Lim if(B ∈ On, B, ∅)))
2		fveq2 2832	. . . . . . 7 ⊢ (B = if(B ∈ On, B, ∅) → (rec(F, A) ‘B) = (rec(F, A) ‘if(B ∈ On, B, ∅)))
3		imaeq2 2603	. . . . . . . 8 ⊢ (B = if(B ∈ On, B, ∅) → (rec(F, A) “ B) = (rec(F, A) “ if(B ∈ On, B, ∅)))
4	3	unieqd 1929	. . . . . . 7 ⊢ (B = if(B ∈ On, B, ∅) → ∪(rec(F, A) “ B) = ∪(rec(F, A) “ if(B ∈ On, B, ∅)))
5	2, 4	cleq12d 1115	. . . . . 6 ⊢ (B = if(B ∈ On, B, ∅) → ((rec(F, A) ‘B) = ∪(rec(F, A) “ B) ↔ (rec(F, A) ‘if(B ∈ On, B, ∅)) = ∪(rec(F, A) “ if(B ∈ On, B, ∅))))
6	1, 5	imbi12d 474	. . . . 5 ⊢ (B = if(B ∈ On, B, ∅) → ((Lim B → (rec(F, A) ‘B) = ∪(rec(F, A) “ B)) ↔ (Lim if(B ∈ On, B, ∅) → (rec(F, A) ‘if(B ∈ On, B, ∅)) = ∪(rec(F, A) “ if(B ∈ On, B, ∅)))))
7		0elon 2277	. . . . . . 7 ⊢ ∅ ∈ On
8	7	elimel 1793	. . . . . 6 ⊢ if(B ∈ On, B, ∅) ∈ On
9	8	rdglim 2981	. . . . 5 ⊢ (Lim if(B ∈ On, B, ∅) → (rec(F, A) ‘if(B ∈ On, B, ∅)) = ∪(rec(F, A) “ if(B ∈ On, B, ∅)))
10	6, 9	dedth 1784	. . . 4 ⊢ (B ∈ On → (Lim B → (rec(F, A) ‘B) = ∪(rec(F, A) “ B)))
11	10	imp 277	. . 3 ⊢ ((B ∈ On ∧ Lim B) → (rec(F, A) ‘B) = ∪(rec(F, A) “ B))
12		limelon 2286	. . 3 ⊢ ((B ∈ C ∧ Lim B) → B ∈ On)
13	11, 12	sylan 343	. 2 ⊢ (((B ∈ C ∧ Lim B) ∧ Lim B) → (rec(F, A) ‘B) = ∪(rec(F, A) “ B))
14	13	anabss3 382	1 ⊢ ((B ∈ C ∧ Lim B) → (rec(F, A) ‘B) = ∪(rec(F, A) “ B))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  ifcif 1776  ∪cuni 1919  Oncon0 2199  Lim wlim 2200   “ cima 2413   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  rdglim2 2987

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

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-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-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  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-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-fv 2438  df-rdg 2970

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