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Theorem rdglim2 8057
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 8051 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))
2 dfima3 5925 . . . . 5 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))}
3 df-rex 3141 . . . . . . 7 (∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ ∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
4 limord 6243 . . . . . . . . . . 11 (Lim 𝐵 → Ord 𝐵)
5 ordelord 6206 . . . . . . . . . . . . 13 ((Ord 𝐵𝑥𝐵) → Ord 𝑥)
65ex 413 . . . . . . . . . . . 12 (Ord 𝐵 → (𝑥𝐵 → Ord 𝑥))
7 vex 3495 . . . . . . . . . . . . 13 𝑥 ∈ V
87elon 6193 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
96, 8syl6ibr 253 . . . . . . . . . . 11 (Ord 𝐵 → (𝑥𝐵𝑥 ∈ On))
104, 9syl 17 . . . . . . . . . 10 (Lim 𝐵 → (𝑥𝐵𝑥 ∈ On))
11 eqcom 2825 . . . . . . . . . . 11 (𝑦 = (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) = 𝑦)
12 rdgfnon 8043 . . . . . . . . . . . 12 rec(𝐹, 𝐴) Fn On
13 fnopfvb 6712 . . . . . . . . . . . 12 ((rec(𝐹, 𝐴) Fn On ∧ 𝑥 ∈ On) → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1412, 13mpan 686 . . . . . . . . . . 11 (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)))
1511, 14syl5bb 284 . . . . . . . . . 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 577 . . . . . . . 8 (Lim 𝐵 → ((𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
1817exbidv 1913 . . . . . . 7 (Lim 𝐵 → (∃𝑥(𝑥𝐵𝑦 = (rec(𝐹, 𝐴)‘𝑥)) ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))))
193, 18syl5rbb 285 . . . . . 6 (Lim 𝐵 → (∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴)) ↔ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)))
2019abbidv 2882 . . . . 5 (Lim 𝐵 → {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ rec(𝐹, 𝐴))} = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
212, 20syl5eq 2865 . . . 4 (Lim 𝐵 → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2221unieqd 4840 . . 3 (Lim 𝐵 (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
2322adantl 482 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴) “ 𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
241, 23eqtrd 2853 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 207  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wrex 3136  cop 4563   cuni 4830  cima 5551  Ord word 6183  Oncon0 6184  Lim wlim 6185   Fn wfn 6343  cfv 6348  reccrdg 8034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-wrecs 7936  df-recs 7997  df-rdg 8035
This theorem is referenced by:  rdglim2a  8058
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