Theorem rdglim2 2987

Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values.

Assertion

Ref	Expression
rdglim2	⊢ ((B ∈ C ∧ Lim B) → (rec(F, A) ‘B) = ∪{y∣∃x ∈ B y = (rec(F, A) ‘x)})

Distinct variable group(s):   x,y,A   x,B,y   x,F,y

Proof of Theorem rdglim2

Step	Hyp	Ref	Expression
1		rdglimt 2986	. 2 ⊢ ((B ∈ C ∧ Lim B) → (rec(F, A) ‘B) = ∪(rec(F, A) “ B))
2		limord 2283	. . . . . . . . . . 11 ⊢ (Lim B → Ord B)
3		ordelord 2221	. . . . . . . . . . . . 13 ⊢ ((Ord B ∧ x ∈ B) → Ord x)
4	3	exp 291	. . . . . . . . . . . 12 ⊢ (Ord B → (x ∈ B → Ord x))
5		visset 1350	. . . . . . . . . . . . 13 ⊢ x ∈ V
6	5	elon 2208	. . . . . . . . . . . 12 ⊢ (x ∈ On ↔ Ord x)
7	4, 6	syl6ibr 186	. . . . . . . . . . 11 ⊢ (Ord B → (x ∈ B → x ∈ On))
8	2, 7	syl 12	. . . . . . . . . 10 ⊢ (Lim B → (x ∈ B → x ∈ On))
9		rdgfnon 2977	. . . . . . . . . . . 12 ⊢ rec(F, A) Fn On
10		visset 1350	. . . . . . . . . . . . 13 ⊢ y ∈ V
11	10	fnfvop 2856	. . . . . . . . . . . 12 ⊢ ((rec(F, A) Fn On ∧ x ∈ On) → ((rec(F, A) ‘x) = y ↔ ⟨x, y⟩ ∈ rec(F, A)))
12	9, 11	mpan 518	. . . . . . . . . . 11 ⊢ (x ∈ On → ((rec(F, A) ‘x) = y ↔ ⟨x, y⟩ ∈ rec(F, A)))
13		cleqcom 1103	. . . . . . . . . . 11 ⊢ (y = (rec(F, A) ‘x) ↔ (rec(F, A) ‘x) = y)
14	12, 13	syl5bb 410	. . . . . . . . . 10 ⊢ (x ∈ On → (y = (rec(F, A) ‘x) ↔ ⟨x, y⟩ ∈ rec(F, A)))
15	8, 14	syl6 23	. . . . . . . . 9 ⊢ (Lim B → (x ∈ B → (y = (rec(F, A) ‘x) ↔ ⟨x, y⟩ ∈ rec(F, A))))
16	15	pm5.32d 491	. . . . . . . 8 ⊢ (Lim B → ((x ∈ B ∧ y = (rec(F, A) ‘x)) ↔ (x ∈ B ∧ ⟨x, y⟩ ∈ rec(F, A))))
17	16	biexdv 936	. . . . . . 7 ⊢ (Lim B → (∃x(x ∈ B ∧ y = (rec(F, A) ‘x)) ↔ ∃x(x ∈ B ∧ ⟨x, y⟩ ∈ rec(F, A))))
18		df-rex 1206	. . . . . . 7 ⊢ (∃x ∈ B y = (rec(F, A) ‘x) ↔ ∃x(x ∈ B ∧ y = (rec(F, A) ‘x)))
19	17, 18	syl5rbb 411	. . . . . 6 ⊢ (Lim B → (∃x(x ∈ B ∧ ⟨x, y⟩ ∈ rec(F, A)) ↔ ∃x ∈ B y = (rec(F, A) ‘x)))
20	19	biabdv 1183	. . . . 5 ⊢ (Lim B → {y∣∃x(x ∈ B ∧ ⟨x, y⟩ ∈ rec(F, A))} = {y∣∃x ∈ B y = (rec(F, A) ‘x)})
21		dfima3 2605	. . . . 5 ⊢ (rec(F, A) “ B) = {y∣∃x(x ∈ B ∧ ⟨x, y⟩ ∈ rec(F, A))}
22	20, 21	syl5eq 1136	. . . 4 ⊢ (Lim B → (rec(F, A) “ B) = {y∣∃x ∈ B y = (rec(F, A) ‘x)})
23	22	unieqd 1929	. . 3 ⊢ (Lim B → ∪(rec(F, A) “ B) = ∪{y∣∃x ∈ B y = (rec(F, A) ‘x)})
24	23	adantl 305	. 2 ⊢ ((B ∈ C ∧ Lim B) → ∪(rec(F, A) “ B) = ∪{y∣∃x ∈ B y = (rec(F, A) ‘x)})
25	1, 24	eqtrd 1128	1 ⊢ ((B ∈ C ∧ Lim B) → (rec(F, A) ‘B) = ∪{y∣∃x ∈ B y = (rec(F, A) ‘x)})

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810  ∪cuni 1919  Ord word 2198  Oncon0 2199  Lim wlim 2200   “ cima 2413   Fn wfn 2417   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  rdglim2a 2988

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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