Theorem rdglim2 8357

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 8351	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵))
2		dfima3 6016	. . . . 5 ⊢ (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3		df-rex 3058	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4		limord 6372	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → Ord 𝐵)
5		ordelord 6333	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → Ord 𝑥)
6	5	ex 412	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → Ord 𝑥))
7		vex 3441	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8	7	elon 6320	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
9	6, 8	imbitrrdi 252	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
10	4, 9	syl 17	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
11		eqcom 2740	. . . . . . . . . . 11 ⊢ (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12		rdgfnon 8343	. . . . . . . . . . . 12 ⊢ rec(𝐹, 𝐴) Fn On
13		fnopfvb 6879	. . . . . . . . . . . 12 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
14	12, 13	mpan 690	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
15	11, 14	bitrid 283	. . . . . . . . . 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 577	. . . . . . . 8 ⊢ (Lim 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
18	17	exbidv 1922	. . . . . . 7 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
19	3, 18	bitr2id 284	. . . . . 6 ⊢ (Lim 𝐵 → (∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
20	19	abbidv 2799	. . . . 5 ⊢ (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
21	2, 20	eqtrid 2780	. . . 4 ⊢ (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
22	21	unieqd 4871	. . 3 ⊢ (Lim 𝐵 → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
23	22	adantl 481	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ (rec(𝐹, 𝐴) “ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
24	1, 23	eqtrd 2768	1 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  ∃wrex 3057  ⟨cop 4581  ∪ cuni 4858   “ cima 5622  Ord word 6310  Oncon0 6311  Lim wlim 6312   Fn wfn 6481  ‘cfv 6486  reccrdg 8334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335

This theorem is referenced by:  rdglim2a  8358