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Theorem rdglim2 8438
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 8432 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))
2 dfima3 6062 . . . . 5 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3 df-rex 3070 . . . . . . 7 (∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4 limord 6424 . . . . . . . . . . 11 (Lim 𝐵 → Ord 𝐵)
5 ordelord 6386 . . . . . . . . . . . . 13 ((Ord 𝐵𝑥𝐵) → Ord 𝑥)
65ex 412 . . . . . . . . . . . 12 (Ord 𝐵 → (𝑥𝐵 → Ord 𝑥))
7 vex 3477 . . . . . . . . . . . . 13 𝑥 ∈ V
87elon 6373 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
96, 8imbitrrdi 251 . . . . . . . . . . 11 (Ord 𝐵 → (𝑥𝐵𝑥 ∈ On))
104, 9syl 17 . . . . . . . . . 10 (Lim 𝐵 → (𝑥𝐵𝑥 ∈ On))
11 eqcom 2738 . . . . . . . . . . 11 (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12 rdgfnon 8424 . . . . . . . . . . . 12 rec(𝐹, 𝐴) Fn On
13 fnopfvb 6945 . . . . . . . . . . . 12 ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1412, 13mpan 687 . . . . . . . . . . 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 576 . . . . . . . 8 (Lim 𝐵 → ((𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
1817exbidv 1923 . . . . . . 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 2800 . . . . 5 (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
212, 20eqtrid 2783 . . . 4 (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2221unieqd 4922 . . 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 2771 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 395   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wrex 3069  cop 4634   cuni 4908  cima 5679  Ord word 6363  Oncon0 6364  Lim wlim 6365   Fn wfn 6538  cfv 6543  reccrdg 8415
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416
This theorem is referenced by:  rdglim2a  8439
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