MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frrlem8 Structured version   Visualization version   GIF version

Theorem frrlem8 8319
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 3483 . . 3 𝑧 ∈ V
21eldm2 5911 . 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 8316 . . . . . . 7 𝐹 = 𝐵
63frrlem1 8312 . . . . . . . 8 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
76unieqi 4918 . . . . . . 7 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
85, 7eqtri 2764 . . . . . 6 𝐹 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
98eleq2i 2832 . . . . 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 4920 . . . . 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 1233 . . . . . . . . . . 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 3483 . . . . . . . . . . . . . 14 𝑤 ∈ V
141, 13opeldm 5917 . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . 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 6672 . . . . . . . . . . . 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 2842 . . . . . . . . . . 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 3246 . . . . . . . . . . 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 4020 . . . . . . . . 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 2180 . . . . . . . . . . . . . 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 2884 . . . . . . . . . . . . . 14 (𝑔𝐵 ↔ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))
2422, 23sylibr 234 . . . . . . . . . . . . 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 4936 . . . . . . . . . . . 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 4024 . . . . . . . . . 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 5912 . . . . . . . . . 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 3993 . . . . . . . 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 1929 . . . . . 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 1929 . . . 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 217 . . 3 (⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
3736exlimiv 1929 . 2 (∃𝑤𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
382, 37sylbi 217 1 (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wss 3950  cop 4631   cuni 4906  dom cdm 5684  cres 5686  Predcpred 6319   Fn wfn 6555  cfv 6560  (class class class)co 7432  frecscfrecs 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ov 7435  df-frecs 8307
This theorem is referenced by:  frrlem12  8323  frrlem13  8324  frrdmcl  8334
  Copyright terms: Public domain W3C validator