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Theorem wfrlem13 7771
Description: Lemma for well-founded recursion. From here through wfrlem16 7774, we aim to prove that dom 𝐹 = 𝐴. We do this by supposing that there is an element 𝑧 of 𝐴 that is not in dom 𝐹. We then define 𝐶 by extending dom 𝐹 with the appropriate value at 𝑧. We then show that 𝑧 cannot be an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), meaning that (𝐴 ∖ dom 𝐹) must be empty, so dom 𝐹 = 𝐴. Here, we show that 𝐶 is a function extending the domain of 𝐹 by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem13.1 𝑅 We 𝐴
wfrlem13.2 𝑅 Se 𝐴
wfrlem13.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13.4 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
Assertion
Ref Expression
wfrlem13 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝑅
Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem13
StepHypRef Expression
1 wfrlem13.1 . . . . 5 𝑅 We 𝐴
2 wfrlem13.2 . . . . 5 𝑅 Se 𝐴
3 wfrlem13.3 . . . . 5 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
41, 2, 3wfrfun 7769 . . . 4 Fun 𝐹
5 vex 3419 . . . . 5 𝑧 ∈ V
6 fvex 6512 . . . . 5 (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
75, 6funsn 6240 . . . 4 Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
84, 7pm3.2i 463 . . 3 (Fun 𝐹 ∧ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
96dmsnop 5912 . . . . 5 dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
109ineq2i 4074 . . . 4 (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∩ {𝑧})
11 eldifn 3995 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
12 disjsn 4521 . . . . 5 ((dom 𝐹 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ dom 𝐹)
1311, 12sylibr 226 . . . 4 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ {𝑧}) = ∅)
1410, 13syl5eq 2827 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = ∅)
15 funun 6233 . . 3 (((Fun 𝐹 ∧ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∧ (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = ∅) → Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
168, 14, 15sylancr 578 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
17 dmun 5629 . . 3 dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
189uneq2i 4026 . . 3 (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
1917, 18eqtri 2803 . 2 dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
20 wfrlem13.4 . . . 4 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
2120fneq1i 6283 . . 3 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn (dom 𝐹 ∪ {𝑧}))
22 df-fn 6191 . . 3 ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn (dom 𝐹 ∪ {𝑧}) ↔ (Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∧ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})))
2321, 22bitri 267 . 2 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∧ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})))
2416, 19, 23sylanblrc 581 1 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  cdif 3827  cun 3828  cin 3829  c0 4179  {csn 4441  cop 4447   Se wse 5364   We wwe 5365  dom cdm 5407  cres 5409  Predcpred 5985  Fun wfun 6182   Fn wfn 6183  cfv 6188  wrecscwrecs 7749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-iota 6152  df-fun 6190  df-fn 6191  df-fv 6196  df-wrecs 7750
This theorem is referenced by:  wfrlem14  7772  wfrlem15  7773
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