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Theorem wfrlem16 7584
Description: Lemma for well-founded 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 wfrdmss 7575), dom 𝐹 = 𝐴. (Contributed by Scott Fenton, 21-Apr-2011.)
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
wfrlem16 dom 𝐹 = 𝐴
Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝑅
Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem16
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem13.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
21wfrdmss 7575 . 2 dom 𝐹𝐴
3 eldifn 3885 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
4 ssun2 3929 . . . . . . . . 9 {𝑧} ⊆ (dom 𝐹 ∪ {𝑧})
5 vsnid 4349 . . . . . . . . 9 𝑧 ∈ {𝑧}
64, 5sselii 3750 . . . . . . . 8 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
7 wfrlem13.4 . . . . . . . . . 10 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
87dmeqi 5464 . . . . . . . . 9 dom 𝐶 = dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9 dmun 5470 . . . . . . . . 9 dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
10 fvex 6343 . . . . . . . . . . 11 (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
1110dmsnop 5752 . . . . . . . . . 10 dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
1211uneq2i 3916 . . . . . . . . 9 (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
138, 9, 123eqtri 2797 . . . . . . . 8 dom 𝐶 = (dom 𝐹 ∪ {𝑧})
146, 13eleqtrri 2849 . . . . . . 7 𝑧 ∈ dom 𝐶
15 wfrlem13.1 . . . . . . . . . . . 12 𝑅 We 𝐴
16 wfrlem13.2 . . . . . . . . . . . 12 𝑅 Se 𝐴
1715, 16, 1, 7wfrlem15 7583 . . . . . . . . . . 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 4604 . . . . . . . . . . 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 df-wrecs 7560 . . . . . . . . . . 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 2793 . . . . . . . . . 10 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2219, 21syl6sseqr 3802 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶𝐹)
23 dmss 5462 . . . . . . . . 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 3752 . . . . . . 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 810 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
2827nrex 3148 . . . 4 ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅
29 df-ne 2944 . . . . 5 ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
30 difss 3889 . . . . . 6 (𝐴 ∖ dom 𝐹) ⊆ 𝐴
3115, 16tz6.26i 5856 . . . . . 6 (((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3230, 31mpan 664 . . . . 5 ((𝐴 ∖ dom 𝐹) ≠ ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3329, 32sylbir 225 . . . 4 (¬ (𝐴 ∖ dom 𝐹) = ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
3428, 33mt3 192 . . 3 (𝐴 ∖ dom 𝐹) = ∅
35 ssdif0 4090 . . 3 (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
3634, 35mpbir 221 . 2 𝐴 ⊆ dom 𝐹
372, 36eqssi 3769 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  cdif 3721  cun 3722  wss 3724  c0 4064  {csn 4317  cop 4323   cuni 4575   Se wse 5207   We wwe 5208  dom cdm 5250  cres 5252  Predcpred 5823   Fn wfn 6027  cfv 6032  wrecscwrecs 7559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-wrecs 7560
This theorem is referenced by:  wfr1  7587  wfr2  7588
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