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Theorem frrlem14 8324
Description: Lemma for well-founded recursion. Finally, we tie all these threads together and show that dom 𝐹 = 𝐴 when given the right 𝑆. Specifically, we prove that there can be no 𝑅-minimal element of (𝐴 ∖ dom 𝐹). (Contributed by Scott Fenton, 7-Dec-2022.)
Hypotheses
Ref Expression
frrlem11.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem11.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem11.3 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
frrlem11.4 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
frrlem12.5 (𝜑𝑅 Fr 𝐴)
frrlem12.6 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
frrlem12.7 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
frrlem13.8 ((𝜑𝑧𝐴) → 𝑆 ∈ V)
frrlem13.9 ((𝜑𝑧𝐴) → 𝑆𝐴)
frrlem14.10 ((𝜑 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
Assertion
Ref Expression
frrlem14 (𝜑 → dom 𝐹 = 𝐴)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧   𝑅,𝑓,𝑥,𝑦,𝑧   𝐵,𝑔,,𝑧   𝑥,𝐹,𝑢,𝑣,𝑧   𝜑,𝑓,𝑧   𝑓,𝐹   𝜑,𝑔,,𝑥,𝑢,𝑣   𝐴,,𝑤,𝑓,𝑦,𝑥   𝑤,𝐺   𝑤,𝑅   𝑦,𝐹   𝑥,𝐵   𝑤,𝐶   𝑤,𝐹   𝜑,𝑤   𝑤,𝑆   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣,𝑢,𝑔)   𝐵(𝑦,𝑤,𝑣,𝑢,𝑓)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,)   𝑅(𝑣,𝑢,𝑔,)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,)   𝐹(𝑔,)   𝐺(𝑣,𝑢,𝑔,)

Proof of Theorem frrlem14
StepHypRef Expression
1 frrlem11.1 . . . 4 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 frrlem11.2 . . . 4 𝐹 = frecs(𝑅, 𝐴, 𝐺)
31, 2frrlem7 8317 . . 3 dom 𝐹𝐴
43a1i 11 . 2 (𝜑 → dom 𝐹𝐴)
5 eldifn 4132 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
65adantl 481 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ 𝑧 ∈ dom 𝐹)
7 frrlem11.3 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
8 frrlem11.4 . . . . . . . . . . . 12 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9 frrlem12.5 . . . . . . . . . . . 12 (𝜑𝑅 Fr 𝐴)
10 frrlem12.6 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
11 frrlem12.7 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
12 frrlem13.8 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → 𝑆 ∈ V)
13 frrlem13.9 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → 𝑆𝐴)
141, 2, 7, 8, 9, 10, 11, 12, 13frrlem13 8323 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐵)
15 elssuni 4937 . . . . . . . . . . 11 (𝐶𝐵𝐶 𝐵)
1614, 15syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 𝐵)
171, 2frrlem5 8315 . . . . . . . . . 10 𝐹 = 𝐵
1816, 17sseqtrrdi 4025 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐹)
19 dmss 5913 . . . . . . . . 9 (𝐶𝐹 → dom 𝐶 ⊆ dom 𝐹)
2018, 19syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → dom 𝐶 ⊆ dom 𝐹)
21 ssun2 4179 . . . . . . . . . . 11 {𝑧} ⊆ (dom (𝐹𝑆) ∪ {𝑧})
22 vsnid 4663 . . . . . . . . . . 11 𝑧 ∈ {𝑧}
2321, 22sselii 3980 . . . . . . . . . 10 𝑧 ∈ (dom (𝐹𝑆) ∪ {𝑧})
248dmeqi 5915 . . . . . . . . . . 11 dom 𝐶 = dom ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
25 dmun 5921 . . . . . . . . . . 11 dom ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
26 ovex 7464 . . . . . . . . . . . . 13 (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
2726dmsnop 6236 . . . . . . . . . . . 12 dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
2827uneq2i 4165 . . . . . . . . . . 11 (dom (𝐹𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹𝑆) ∪ {𝑧})
2924, 25, 283eqtri 2769 . . . . . . . . . 10 dom 𝐶 = (dom (𝐹𝑆) ∪ {𝑧})
3023, 29eleqtrri 2840 . . . . . . . . 9 𝑧 ∈ dom 𝐶
3130a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ dom 𝐶)
3220, 31sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ dom 𝐹)
3332expr 456 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → 𝑧 ∈ dom 𝐹))
346, 33mtod 198 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3534nrexdv 3149 . . . 4 (𝜑 → ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
36 df-ne 2941 . . . . . 6 ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
37 frrlem14.10 . . . . . 6 ((𝜑 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3836, 37sylan2br 595 . . . . 5 ((𝜑 ∧ ¬ (𝐴 ∖ dom 𝐹) = ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3938ex 412 . . . 4 (𝜑 → (¬ (𝐴 ∖ dom 𝐹) = ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅))
4035, 39mt3d 148 . . 3 (𝜑 → (𝐴 ∖ dom 𝐹) = ∅)
41 ssdif0 4366 . . 3 (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
4240, 41sylibr 234 . 2 (𝜑𝐴 ⊆ dom 𝐹)
434, 42eqssd 4001 1 (𝜑 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  wss 3951  c0 4333  {csn 4626  cop 4632   cuni 4907   class class class wbr 5143   Fr wfr 5634  dom cdm 5685  cres 5687  Predcpred 6320   Fn wfn 6556  cfv 6561  (class class class)co 7431  frecscfrecs 8305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-fr 5637  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-frecs 8306
This theorem is referenced by:  fpr1  8328  frr1  9799
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