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Theorem wfrlem16OLD 8363
Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. If 𝑧 is 𝑅 minimal in (𝐴 ∖ dom 𝐹), then 𝐶 is acceptable and thus a subset of 𝐹, but dom 𝐶 is bigger than dom 𝐹. Thus, 𝑧 cannot be minimal, so (𝐴 ∖ dom 𝐹) must be empty, and (due to wfrdmssOLD 8354), dom 𝐹 = 𝐴. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem13OLD.1 𝑅 We 𝐴
wfrlem13OLD.2 𝑅 Se 𝐴
wfrlem13OLD.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13OLD.4 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
Assertion
Ref Expression
wfrlem16OLD dom 𝐹 = 𝐴
Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝑅
Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem16OLD
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem13OLD.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
21wfrdmssOLD 8354 . 2 dom 𝐹𝐴
3 eldifn 4142 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
4 ssun2 4189 . . . . . . . . 9 {𝑧} ⊆ (dom 𝐹 ∪ {𝑧})
5 vsnid 4668 . . . . . . . . 9 𝑧 ∈ {𝑧}
64, 5sselii 3992 . . . . . . . 8 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
7 wfrlem13OLD.4 . . . . . . . . . 10 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
87dmeqi 5918 . . . . . . . . 9 dom 𝐶 = dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9 dmun 5924 . . . . . . . . 9 dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
10 fvex 6920 . . . . . . . . . . 11 (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
1110dmsnop 6238 . . . . . . . . . 10 dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
1211uneq2i 4175 . . . . . . . . 9 (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
138, 9, 123eqtri 2767 . . . . . . . 8 dom 𝐶 = (dom 𝐹 ∪ {𝑧})
146, 13eleqtrri 2838 . . . . . . 7 𝑧 ∈ dom 𝐶
15 wfrlem13OLD.1 . . . . . . . . . . . 12 𝑅 We 𝐴
16 wfrlem13OLD.2 . . . . . . . . . . . 12 𝑅 Se 𝐴
1715, 16, 1, 7wfrlem15OLD 8362 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
18 elssuni 4942 . . . . . . . . . . 11 (𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝐶 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
1917, 18syl 17 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
20 dfwrecsOLD 8337 . . . . . . . . . . 11 wrecs(𝑅, 𝐴, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
211, 20eqtri 2763 . . . . . . . . . 10 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2219, 21sseqtrrdi 4047 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶𝐹)
23 dmss 5916 . . . . . . . . 9 (𝐶𝐹 → dom 𝐶 ⊆ dom 𝐹)
2422, 23syl 17 . . . . . . . 8 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐶 ⊆ dom 𝐹)
2524sseld 3994 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑧 ∈ dom 𝐶𝑧 ∈ dom 𝐹))
2614, 25mpi 20 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝑧 ∈ dom 𝐹)
273, 26mtand 816 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
2827nrex 3072 . . . 4 ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅
29 df-ne 2939 . . . . 5 ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
30 difss 4146 . . . . . 6 (𝐴 ∖ dom 𝐹) ⊆ 𝐴
3115, 16tz6.26i 6372 . . . . . 6 (((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3230, 31mpan 690 . . . . 5 ((𝐴 ∖ dom 𝐹) ≠ ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3329, 32sylbir 235 . . . 4 (¬ (𝐴 ∖ dom 𝐹) = ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3428, 33mt3 201 . . 3 (𝐴 ∖ dom 𝐹) = ∅
35 ssdif0 4372 . . 3 (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
3634, 35mpbir 231 . 2 𝐴 ⊆ dom 𝐹
372, 36eqssi 4012 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  cdif 3960  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637   cuni 4912   Se wse 5639   We wwe 5640  dom cdm 5689  cres 5691  Predcpred 6322   Fn wfn 6558  cfv 6563  wrecscwrecs 8335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336
This theorem is referenced by:  wfr1OLD  8366  wfr2OLD  8367
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