Theorem rdglim2 8368

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 8362	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵))
2		dfima3 6022	. . . . 5 ⊢ (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3		df-rex 3065	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4		limord 6378	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → Ord 𝐵)
5		ordelord 6339	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → Ord 𝑥)
6	5	ex 413	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → Ord 𝑥))
7		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8	7	elon 6326	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
9	6, 8	imbitrrdi 253	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
10	4, 9	syl 17	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
11		eqcom 2747	. . . . . . . . . . 11 ⊢ (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12		rdgfnon 8354	. . . . . . . . . . . 12 ⊢ rec(𝐹, 𝐴) Fn On
13		fnopfvb 6885	. . . . . . . . . . . 12 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
14	12, 13	mpan 696	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
15	11, 14	bitrid 284	. . . . . . . . . 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 582	. . . . . . . 8 ⊢ (Lim 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
18	17	exbidv 1928	. . . . . . 7 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
19	3, 18	bitr2id 285	. . . . . 6 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
20	19	abbidv 2806	. . . . 5 ⊢ (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
21	2, 20	eqtrid 2787	. . . 4 ⊢ (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
22	21	unieqd 4858	. . 3 ⊢ (Lim 𝐵 → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
23	22	adantl 482	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
24	1, 23	eqtrd 2775	1 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718  ∃wrex 3064  ⟨cop 4568  ∪ cuni 4845   “ cima 5628  Ord word 6316  Oncon0 6317  Lim wlim 6318   Fn wfn 6487  ‘cfv 6492  reccrdg 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346

This theorem is referenced by:  rdglim2a  8369