Theorem rdglim2 8422

Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)

Proof of Theorem rdglim2

Step	Hyp	Ref	Expression
1		rdglim 8416	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵))
2		dfima3 6059	. . . . 5 ⊢ (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3		df-rex 3087	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4		limord 6419	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → Ord 𝐵)
5		ordelord 6379	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → Ord 𝑥)
6	5	ex 418	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → Ord 𝑥))
7		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8	7	elon 6366	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
9	6, 8	imbitrrdi 255	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
10	4, 9	syl 18	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
11		eqcom 2767	. . . . . . . . . . 11 ⊢ (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12		rdgfnon 8408	. . . . . . . . . . . 12 ⊢ rec(𝐹, 𝐴) Fn On
13		fnopfvb 6930	. . . . . . . . . . . 12 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
14	12, 13	mpan 703	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
15	11, 14	bitrid 286	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
16	10, 15	syl6 36	. . . . . . . . 9 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
17	16	pm5.32d 588	. . . . . . . 8 ⊢ (Lim 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
18	17	exbidv 1954	. . . . . . 7 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
19	3, 18	bitr2id 287	. . . . . 6 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
20	19	abbidv 2826	. . . . 5 ⊢ (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
21	2, 20	eqtrid 2807	. . . 4 ⊢ (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
22	21	unieqd 4880	. . 3 ⊢ (Lim 𝐵 → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
23	22	adantl 487	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
24	1, 23	eqtrd 2795	1 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086  ⟨cop 4590  ∪ cuni 4867   “ cima 5658  Ord word 6356  Oncon0 6357  Lim wlim 6358   Fn wfn 6528  ‘cfv 6533  reccrdg 8399

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400

This theorem is used by:  rdglim2a  8423