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Theorem frrlem8 8276
Description: Lemma for well-founded recursion. dom 𝐹 is closed under predecessor classes. (Contributed by Scott Fenton, 6-Dec-2022.)
Hypotheses
Ref Expression
frrlem5.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem5.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frrlem8 (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑦,𝐹   𝑧,𝐴,𝑓,𝑥,𝑦   𝑧,𝑅   𝑧,𝐺
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑓)   𝐹(𝑥,𝑧,𝑓)

Proof of Theorem frrlem8
Dummy variables 𝑔 𝑎 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3472 . . 3 𝑧 ∈ V
21eldm2 5894 . 2 (𝑧 ∈ dom 𝐹 ↔ ∃𝑤𝑧, 𝑤⟩ ∈ 𝐹)
3 frrlem5.1 . . . . . . . 8 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
4 frrlem5.2 . . . . . . . 8 𝐹 = frecs(𝑅, 𝐴, 𝐺)
53, 4frrlem5 8273 . . . . . . 7 𝐹 = 𝐵
63frrlem1 8269 . . . . . . . 8 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
76unieqi 4914 . . . . . . 7 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
85, 7eqtri 2754 . . . . . 6 𝐹 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
98eleq2i 2819 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))})
10 eluniab 4916 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))} ↔ ∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))))
119, 10bitri 275 . . . 4 (⟨𝑧, 𝑤⟩ ∈ 𝐹 ↔ ∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))))
12 simpr2r 1230 . . . . . . . . . . 11 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎)
13 vex 3472 . . . . . . . . . . . . . 14 𝑤 ∈ V
141, 13opeldm 5900 . . . . . . . . . . . . 13 (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧 ∈ dom 𝑔)
1514adantr 480 . . . . . . . . . . . 12 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧 ∈ dom 𝑔)
16 simpr1 1191 . . . . . . . . . . . . 13 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 Fn 𝑎)
1716fndmd 6647 . . . . . . . . . . . 12 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → dom 𝑔 = 𝑎)
1815, 17eleqtrd 2829 . . . . . . . . . . 11 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧𝑎)
19 rsp 3238 . . . . . . . . . . 11 (∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎 → (𝑧𝑎 → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎))
2012, 18, 19sylc 65 . . . . . . . . . 10 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎)
2120, 17sseqtrrd 4018 . . . . . . . . 9 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝑔)
22 19.8a 2166 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))
236eqabri 2871 . . . . . . . . . . . . . 14 (𝑔𝐵 ↔ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))
2422, 23sylibr 233 . . . . . . . . . . . . 13 ((𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝑔𝐵)
2524adantl 481 . . . . . . . . . . . 12 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔𝐵)
26 elssuni 4934 . . . . . . . . . . . 12 (𝑔𝐵𝑔 𝐵)
2725, 26syl 17 . . . . . . . . . . 11 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 𝐵)
2827, 5sseqtrrdi 4028 . . . . . . . . . 10 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔𝐹)
29 dmss 5895 . . . . . . . . . 10 (𝑔𝐹 → dom 𝑔 ⊆ dom 𝐹)
3028, 29syl 17 . . . . . . . . 9 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → dom 𝑔 ⊆ dom 𝐹)
3121, 30sstrd 3987 . . . . . . . 8 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
3231expcom 413 . . . . . . 7 ((𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹))
3332exlimiv 1925 . . . . . 6 (∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹))
3433impcom 407 . . . . 5 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
3534exlimiv 1925 . . . 4 (∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
3611, 35sylbi 216 . . 3 (⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
3736exlimiv 1925 . 2 (∃𝑤𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
382, 37sylbi 216 1 (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wral 3055  wss 3943  cop 4629   cuni 4902  dom cdm 5669  cres 5671  Predcpred 6292   Fn wfn 6531  cfv 6536  (class class class)co 7404  frecscfrecs 8263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7407  df-frecs 8264
This theorem is referenced by:  frrlem12  8280  frrlem13  8281  frrdmcl  8291
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