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Theorem rdglim2 8419
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 8413 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))
2 dfima3 6055 . . . . 5 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3 df-rex 3072 . . . . . . 7 (∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4 limord 6416 . . . . . . . . . . 11 (Lim 𝐵 → Ord 𝐵)
5 ordelord 6378 . . . . . . . . . . . . 13 ((Ord 𝐵𝑥𝐵) → Ord 𝑥)
65ex 414 . . . . . . . . . . . 12 (Ord 𝐵 → (𝑥𝐵 → Ord 𝑥))
7 vex 3479 . . . . . . . . . . . . 13 𝑥 ∈ V
87elon 6365 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
96, 8syl6ibr 252 . . . . . . . . . . 11 (Ord 𝐵 → (𝑥𝐵𝑥 ∈ On))
104, 9syl 17 . . . . . . . . . 10 (Lim 𝐵 → (𝑥𝐵𝑥 ∈ On))
11 eqcom 2740 . . . . . . . . . . 11 (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12 rdgfnon 8405 . . . . . . . . . . . 12 rec(𝐹, 𝐴) Fn On
13 fnopfvb 6935 . . . . . . . . . . . 12 ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1412, 13mpan 689 . . . . . . . . . . 11 (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1511, 14bitrid 283 . . . . . . . . . 10 (𝑥 ∈ On → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1610, 15syl6 35 . . . . . . . . 9 (Lim 𝐵 → (𝑥𝐵 → (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
1716pm5.32d 578 . . . . . . . 8 (Lim 𝐵 → ((𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
1817exbidv 1925 . . . . . . 7 (Lim 𝐵 → (∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
193, 18bitr2id 284 . . . . . 6 (Lim 𝐵 → (∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
2019abbidv 2802 . . . . 5 (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
212, 20eqtrid 2785 . . . 4 (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2221unieqd 4918 . . 3 (Lim 𝐵 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2322adantl 483 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
241, 23eqtrd 2773 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 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3071  cop 4630   cuni 4904  cima 5675  Ord word 6355  Oncon0 6356  Lim wlim 6357   Fn wfn 6530  cfv 6535  reccrdg 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397
This theorem is referenced by:  rdglim2a  8420
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