MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfrlem16OLD Structured version   Visualization version   GIF version

Theorem wfrlem16OLD 8225
Description: 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 8216), dom 𝐹 = 𝐴. Obsolete as of 18-Nov-2024. (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 8216 . 2 dom 𝐹𝐴
3 eldifn 4074 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
4 ssun2 4120 . . . . . . . . 9 {𝑧} ⊆ (dom 𝐹 ∪ {𝑧})
5 vsnid 4610 . . . . . . . . 9 𝑧 ∈ {𝑧}
64, 5sselii 3929 . . . . . . . 8 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
7 wfrlem13OLD.4 . . . . . . . . . 10 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
87dmeqi 5846 . . . . . . . . 9 dom 𝐶 = dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9 dmun 5852 . . . . . . . . 9 dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
10 fvex 6838 . . . . . . . . . . 11 (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
1110dmsnop 6154 . . . . . . . . . 10 dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
1211uneq2i 4107 . . . . . . . . 9 (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
138, 9, 123eqtri 2768 . . . . . . . 8 dom 𝐶 = (dom 𝐹 ∪ {𝑧})
146, 13eleqtrri 2836 . . . . . . 7 𝑧 ∈ dom 𝐶
15 wfrlem13OLD.1 . . . . . . . . . . . 12 𝑅 We 𝐴
16 wfrlem13OLD.2 . . . . . . . . . . . 12 𝑅 Se 𝐴
1715, 16, 1, 7wfrlem15OLD 8224 . . . . . . . . . . 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 4885 . . . . . . . . . . 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 8199 . . . . . . . . . . 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 2764 . . . . . . . . . 10 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2219, 21sseqtrrdi 3983 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶𝐹)
23 dmss 5844 . . . . . . . . 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 3931 . . . . . . 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 813 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
2827nrex 3074 . . . 4 ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅
29 df-ne 2941 . . . . 5 ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
30 difss 4078 . . . . . 6 (𝐴 ∖ dom 𝐹) ⊆ 𝐴
3115, 16tz6.26i 6288 . . . . . 6 (((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3230, 31mpan 687 . . . . 5 ((𝐴 ∖ dom 𝐹) ≠ ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3329, 32sylbir 234 . . . 4 (¬ (𝐴 ∖ dom 𝐹) = ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3428, 33mt3 200 . . 3 (𝐴 ∖ dom 𝐹) = ∅
35 ssdif0 4310 . . 3 (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
3634, 35mpbir 230 . 2 𝐴 ⊆ dom 𝐹
372, 36eqssi 3948 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  {cab 2713  wne 2940  wral 3061  wrex 3070  cdif 3895  cun 3896  wss 3898  c0 4269  {csn 4573  cop 4579   cuni 4852   Se wse 5573   We wwe 5574  dom cdm 5620  cres 5622  Predcpred 6237   Fn wfn 6474  cfv 6479  wrecscwrecs 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-2nd 7900  df-frecs 8167  df-wrecs 8198
This theorem is referenced by:  wfr1OLD  8228  wfr2OLD  8229
  Copyright terms: Public domain W3C validator