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Theorem frrlem14 8286
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 8279 . . 3 dom 𝐹𝐴
43a1i 11 . 2 (𝜑 → dom 𝐹𝐴)
5 eldifn 4126 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
65adantl 480 . . . . . 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 8285 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐵)
15 elssuni 4940 . . . . . . . . . . 11 (𝐶𝐵𝐶 𝐵)
1614, 15syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 𝐵)
171, 2frrlem5 8277 . . . . . . . . . 10 𝐹 = 𝐵
1816, 17sseqtrrdi 4032 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐹)
19 dmss 5901 . . . . . . . . 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 4172 . . . . . . . . . . 11 {𝑧} ⊆ (dom (𝐹𝑆) ∪ {𝑧})
22 vsnid 4664 . . . . . . . . . . 11 𝑧 ∈ {𝑧}
2321, 22sselii 3978 . . . . . . . . . 10 𝑧 ∈ (dom (𝐹𝑆) ∪ {𝑧})
248dmeqi 5903 . . . . . . . . . . 11 dom 𝐶 = dom ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
25 dmun 5909 . . . . . . . . . . 11 dom ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
26 ovex 7444 . . . . . . . . . . . . 13 (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
2726dmsnop 6214 . . . . . . . . . . . 12 dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
2827uneq2i 4159 . . . . . . . . . . 11 (dom (𝐹𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹𝑆) ∪ {𝑧})
2924, 25, 283eqtri 2762 . . . . . . . . . 10 dom 𝐶 = (dom (𝐹𝑆) ∪ {𝑧})
3023, 29eleqtrri 2830 . . . . . . . . 9 𝑧 ∈ dom 𝐶
3130a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ dom 𝐶)
3220, 31sseldd 3982 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ dom 𝐹)
3332expr 455 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → 𝑧 ∈ dom 𝐹))
346, 33mtod 197 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3534nrexdv 3147 . . . 4 (𝜑 → ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
36 df-ne 2939 . . . . . 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 593 . . . . 5 ((𝜑 ∧ ¬ (𝐴 ∖ dom 𝐹) = ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3938ex 411 . . . 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 4362 . . 3 (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
4240, 41sylibr 233 . 2 (𝜑𝐴 ⊆ dom 𝐹)
434, 42eqssd 3998 1 (𝜑 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1085   = wceq 1539  wex 1779  wcel 2104  {cab 2707  wne 2938  wral 3059  wrex 3068  Vcvv 3472  cdif 3944  cun 3945  wss 3947  c0 4321  {csn 4627  cop 4633   cuni 4907   class class class wbr 5147   Fr wfr 5627  dom cdm 5675  cres 5677  Predcpred 6298   Fn wfn 6537  cfv 6542  (class class class)co 7411  frecscfrecs 8267
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-fr 5630  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-frecs 8268
This theorem is referenced by:  fpr1  8290  frr1  9756
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