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Theorem frrlem8 8303
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 3475 . . 3 𝑧 ∈ V
21eldm2 5906 . 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 8300 . . . . . . 7 𝐹 = 𝐵
63frrlem1 8296 . . . . . . . 8 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
76unieqi 4922 . . . . . . 7 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
85, 7eqtri 2755 . . . . . 6 𝐹 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
98eleq2i 2820 . . . . 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 4924 . . . . 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 274 . . . 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 3475 . . . . . . . . . . . . . 14 𝑤 ∈ V
141, 13opeldm 5912 . . . . . . . . . . . . 13 (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧 ∈ dom 𝑔)
1514adantr 479 . . . . . . . . . . . 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 6662 . . . . . . . . . . . 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 2830 . . . . . . . . . . 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 3240 . . . . . . . . . . 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 4021 . . . . . . . . 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 2169 . . . . . . . . . . . . . 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 2872 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 4942 . . . . . . . . . . . 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 4031 . . . . . . . . . 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 5907 . . . . . . . . . 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 3990 . . . . . . . 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 412 . . . . . . 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 406 . . . . 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 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2704  wral 3057  wss 3947  cop 4636   cuni 4910  dom cdm 5680  cres 5682  Predcpred 6307   Fn wfn 6546  cfv 6551  (class class class)co 7424  frecscfrecs 8290
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 2166  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-ov 7427  df-frecs 8291
This theorem is referenced by:  frrlem12  8307  frrlem13  8308  frrdmcl  8318
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