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Theorem frrlem8 33137
Description: Lemma for 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 3474 . . 3 𝑧 ∈ V
21eldm2 5743 . 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 33134 . . . . . . 7 𝐹 = 𝐵
63frrlem1 33130 . . . . . . . 8 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
76unieqi 4824 . . . . . . 7 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
85, 7eqtri 2844 . . . . . 6 𝐹 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
98eleq2i 2903 . . . . 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 4826 . . . . 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 278 . . . 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 3474 . . . . . . . . . . . . . 14 𝑤 ∈ V
141, 13opeldm 5749 . . . . . . . . . . . . 13 (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧 ∈ dom 𝑔)
1514adantr 484 . . . . . . . . . . . 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)
17 fndm 6428 . . . . . . . . . . . . 13 (𝑔 Fn 𝑎 → dom 𝑔 = 𝑎)
1816, 17syl 17 . . . . . . . . . . . 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)
1915, 18eleqtrd 2914 . . . . . . . . . . 11 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧𝑎)
20 rsp 3193 . . . . . . . . . . 11 (∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎 → (𝑧𝑎 → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎))
2112, 19, 20sylc 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)
2221, 18sseqtrrd 3984 . . . . . . . . 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)
23 19.8a 2181 . . . . . . . . . . . . . 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)))))
246abeq2i 2947 . . . . . . . . . . . . . 14 (𝑔𝐵 ↔ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))
2523, 24sylibr 237 . . . . . . . . . . . . 13 ((𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝑔𝐵)
2625adantl 485 . . . . . . . . . . . 12 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔𝐵)
27 elssuni 4841 . . . . . . . . . . . 12 (𝑔𝐵𝑔 𝐵)
2826, 27syl 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)
2928, 5sseqtrrdi 3994 . . . . . . . . . 10 ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎𝐴 ∧ ∀𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧𝑎 (𝑔𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔𝐹)
30 dmss 5744 . . . . . . . . . 10 (𝑔𝐹 → dom 𝑔 ⊆ dom 𝐹)
3129, 30syl 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)
3222, 31sstrd 3953 . . . . . . . 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)
3332expcom 417 . . . . . . 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))
3433exlimiv 1932 . . . . . 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))
3534impcom 411 . . . . 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)
3635exlimiv 1932 . . . 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)
3711, 36sylbi 220 . . 3 (⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
3837exlimiv 1932 . 2 (∃𝑤𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
392, 38sylbi 220 1 (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2799  wral 3126  wss 3910  cop 4546   cuni 4811  dom cdm 5528  cres 5530  Predcpred 6120   Fn wfn 6323  cfv 6328  (class class class)co 7130  frecscfrecs 33124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-ov 7133  df-frecs 33125
This theorem is referenced by:  frrlem12  33141  frrlem13  33142
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