Theorem rdglim2 8398

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 8392	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵))
2		dfima3 6049	. . . . 5 ⊢ (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3		df-rex 3086	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4		limord 6403	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → Ord 𝐵)
5		ordelord 6364	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → Ord 𝑥)
6	5	ex 416	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → Ord 𝑥))
7		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8	7	elon 6351	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
9	6, 8	imbitrrdi 254	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
10	4, 9	syl 17	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
11		eqcom 2768	. . . . . . . . . . 11 ⊢ (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12		rdgfnon 8384	. . . . . . . . . . . 12 ⊢ rec(𝐹, 𝐴) Fn On
13		fnopfvb 6914	. . . . . . . . . . . 12 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
14	12, 13	mpan 700	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
15	11, 14	bitrid 285	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
16	10, 15	syl6 35	. . . . . . . . 9 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
17	16	pm5.32d 585	. . . . . . . 8 ⊢ (Lim 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
18	17	exbidv 1940	. . . . . . 7 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
19	3, 18	bitr2id 286	. . . . . 6 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
20	19	abbidv 2827	. . . . 5 ⊢ (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
21	2, 20	eqtrid 2808	. . . 4 ⊢ (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
22	21	unieqd 4877	. . 3 ⊢ (Lim 𝐵 → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
23	22	adantl 485	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
24	1, 23	eqtrd 2796	1 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∃wrex 3085  ⟨cop 4587  ∪ cuni 4864   “ cima 5648  Ord word 6341  Oncon0 6342  Lim wlim 6343   Fn wfn 6512  ‘cfv 6517  reccrdg 8375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376

This theorem is referenced by:  rdglim2a  8399