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Theorem frrlem11 8231
Description: Lemma for well-founded recursion. For the next several theorems we will be aiming to prove that dom 𝐹 = 𝐴. To do this, we set up a function 𝐶 that supposedly contains an element of 𝐴 that is not in dom 𝐹 and we show that the element must be in dom 𝐹. Our choice of what to restrict 𝐹 to depends on if we assume partial orders or the axiom of infinity. To begin with, we establish the functionality of 𝐶. (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)))⟩})
Assertion
Ref Expression
frrlem11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
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
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣,𝑢,𝑔,)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑓)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,)   𝑅(𝑣,𝑢,𝑔,)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,)   𝐹(𝑦,𝑔,)   𝐺(𝑣,𝑢,𝑔,)

Proof of Theorem frrlem11
StepHypRef Expression
1 frrlem11.1 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 frrlem11.2 . . . . . . 7 𝐹 = frecs(𝑅, 𝐴, 𝐺)
3 frrlem11.3 . . . . . . 7 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
41, 2, 3frrlem9 8229 . . . . . 6 (𝜑 → Fun 𝐹)
54funresd 6548 . . . . 5 (𝜑 → Fun (𝐹𝑆))
6 dmres 5963 . . . . . 6 dom (𝐹𝑆) = (𝑆 ∩ dom 𝐹)
7 df-fn 6503 . . . . . 6 ((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ (Fun (𝐹𝑆) ∧ dom (𝐹𝑆) = (𝑆 ∩ dom 𝐹)))
86, 7mpbiran2 709 . . . . 5 ((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ Fun (𝐹𝑆))
95, 8sylibr 233 . . . 4 (𝜑 → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
10 vex 3451 . . . . 5 𝑧 ∈ V
11 ovex 7394 . . . . 5 (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
1210, 11fnsn 6563 . . . 4 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}
139, 12jctir 522 . . 3 (𝜑 → ((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}))
14 eldifn 4091 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
15 elinel2 4160 . . . . 5 (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹)
1614, 15nsyl 140 . . . 4 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
17 disjsn 4676 . . . 4 (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
1816, 17sylibr 233 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
19 fnun 6618 . . 3 ((((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}) ∧ ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
2013, 18, 19syl2an 597 . 2 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
21 frrlem11.4 . . 3 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
2221fneq1i 6603 . 2 (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
2320, 22sylibr 233 1 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  cdif 3911  cun 3912  cin 3913  wss 3914  c0 4286  {csn 4590  cop 4596   class class class wbr 5109  dom cdm 5637  cres 5639  Predcpred 6256  Fun wfun 6494   Fn wfn 6495  cfv 6500  (class class class)co 7361  frecscfrecs 8215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-ov 7364  df-frecs 8216
This theorem is referenced by:  frrlem12  8232  frrlem13  8233
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