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Theorem tfrlemi14d 6386
Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
Hypotheses
Ref Expression
tfrlemi14d.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemi14d.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
Assertion
Ref Expression
tfrlemi14d (𝜑 → dom recs(𝐹) = On)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝑓,𝐹,𝑥,𝑦   𝜑,𝑓,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem tfrlemi14d
Dummy variables 𝑔 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlemi14d.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem8 6371 . . 3 Ord dom recs(𝐹)
3 ordsson 4524 . . 3 (Ord dom recs(𝐹) → dom recs(𝐹) ⊆ On)
42, 3mp1i 10 . 2 (𝜑 → dom recs(𝐹) ⊆ On)
5 tfrlemi14d.2 . . . . . . . 8 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
61, 5tfrlemi1 6385 . . . . . . 7 ((𝜑𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
75ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
8 simplr 528 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑧 ∈ On)
9 simprl 529 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑔 Fn 𝑧)
10 fneq2 5343 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
11 raleq 2690 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1210, 11anbi12d 473 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1312rspcev 2864 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1413adantll 476 . . . . . . . . . 10 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
15 vex 2763 . . . . . . . . . . 11 𝑔 ∈ V
161, 15tfrlem3a 6363 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1714, 16sylibr 134 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑔𝐴)
181, 7, 8, 9, 17tfrlemisucaccv 6378 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
19 vex 2763 . . . . . . . . . . . 12 𝑧 ∈ V
205tfrlem3-2d 6365 . . . . . . . . . . . . 13 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
2120simprd 114 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑔) ∈ V)
22 opexg 4257 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
2319, 21, 22sylancr 414 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
24 snidg 3647 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
25 elun2 3327 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩} → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
2623, 24, 253syl 17 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
2726ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
28 opeldmg 4867 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
2919, 21, 28sylancr 414 . . . . . . . . . 10 (𝜑 → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3029ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3127, 30mpd 13 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
32 dmeq 4862 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3332eleq2d 2263 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3433rspcev 2864 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
3518, 31, 34syl2anc 411 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
366, 35exlimddv 1910 . . . . . 6 ((𝜑𝑧 ∈ On) → ∃𝐴 𝑧 ∈ dom )
37 eliun 3916 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
3836, 37sylibr 134 . . . . 5 ((𝜑𝑧 ∈ On) → 𝑧 𝐴 dom )
3938ex 115 . . . 4 (𝜑 → (𝑧 ∈ On → 𝑧 𝐴 dom ))
4039ssrdv 3185 . . 3 (𝜑 → On ⊆ 𝐴 dom )
411recsfval 6368 . . . . 5 recs(𝐹) = 𝐴
4241dmeqi 4863 . . . 4 dom recs(𝐹) = dom 𝐴
43 dmuni 4872 . . . 4 dom 𝐴 = 𝐴 dom
4442, 43eqtri 2214 . . 3 dom recs(𝐹) = 𝐴 dom
4540, 44sseqtrrdi 3228 . 2 (𝜑 → On ⊆ dom recs(𝐹))
464, 45eqssd 3196 1 (𝜑 → dom recs(𝐹) = On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362   = wceq 1364  wcel 2164  {cab 2179  wral 2472  wrex 2473  Vcvv 2760  cun 3151  wss 3153  {csn 3618  cop 3621   cuni 3835   ciun 3912  Ord word 4393  Oncon0 4394  dom cdm 4659  cres 4661  Fun wfun 5248   Fn wfn 5249  cfv 5254  recscrecs 6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358
This theorem is referenced by:  tfri1d  6388
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