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Theorem rdglim2 7415
 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 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = {𝑦 ∣ ∃𝑥𝐵 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
StepHypRef Expression
1 rdglim 7409 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))
2 dfima3 5388 . . . . 5 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3 df-rex 2902 . . . . . . 7 (∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4 limord 5701 . . . . . . . . . . 11 (Lim 𝐵 → Ord 𝐵)
5 ordelord 5662 . . . . . . . . . . . . 13 ((Ord 𝐵𝑥𝐵) → Ord 𝑥)
65ex 449 . . . . . . . . . . . 12 (Ord 𝐵 → (𝑥𝐵 → Ord 𝑥))
7 vex 3176 . . . . . . . . . . . . 13 𝑥 ∈ V
87elon 5649 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
96, 8syl6ibr 241 . . . . . . . . . . 11 (Ord 𝐵 → (𝑥𝐵𝑥 ∈ On))
104, 9syl 17 . . . . . . . . . 10 (Lim 𝐵 → (𝑥𝐵𝑥 ∈ On))
11 eqcom 2617 . . . . . . . . . . 11 (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12 rdgfnon 7401 . . . . . . . . . . . 12 rec(𝐹, 𝐴) Fn On
13 fnopfvb 6147 . . . . . . . . . . . 12 ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1412, 13mpan 702 . . . . . . . . . . 11 (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1511, 14syl5bb 271 . . . . . . . . . 10 (𝑥 ∈ On → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1610, 15syl6 34 . . . . . . . . 9 (Lim 𝐵 → (𝑥𝐵 → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
1716pm5.32d 669 . . . . . . . 8 (Lim 𝐵 → ((𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
1817exbidv 1837 . . . . . . 7 (Lim 𝐵 → (∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
193, 18syl5rbb 272 . . . . . 6 (Lim 𝐵 → (∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
2019abbidv 2728 . . . . 5 (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
212, 20syl5eq 2656 . . . 4 (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2221unieqd 4382 . . 3 (Lim 𝐵 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2322adantl 481 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
241, 23eqtrd 2644 1 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ⟨cop 4131  ∪ cuni 4372   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641   Fn wfn 5799  ‘cfv 5804  reccrdg 7392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393 This theorem is referenced by:  rdglim2a  7416
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