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Theorem tfrlemi14d 5977
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 5964 . . 3 Ord dom recs(𝐹)
3 ordsson 4245 . . 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 5976 . . . . . . 7 ((𝜑𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
75ad2antrr 465 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
8 simplr 490 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑧 ∈ On)
9 simprl 491 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑔 Fn 𝑧)
10 fneq2 5015 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
11 raleq 2522 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1210, 11anbi12d 450 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1312rspcev 2673 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1413adantll 453 . . . . . . . . . 10 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
15 vex 2577 . . . . . . . . . . 11 𝑔 ∈ V
161, 15tfrlem3a 5955 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1714, 16sylibr 141 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑔𝐴)
181, 7, 8, 9, 17tfrlemisucaccv 5969 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
19 vex 2577 . . . . . . . . . . . 12 𝑧 ∈ V
205tfrlem3-2d 5958 . . . . . . . . . . . . 13 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
2120simprd 111 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑔) ∈ V)
22 opexg 3991 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
2319, 21, 22sylancr 399 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
24 snidg 3427 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
25 elun2 3138 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩} → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
2623, 24, 253syl 17 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
2726ad2antrr 465 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
28 opeldmg 4567 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
2919, 21, 28sylancr 399 . . . . . . . . . 10 (𝜑 → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3029ad2antrr 465 . . . . . . . . 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 4562 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3332eleq2d 2123 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3433rspcev 2673 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
3518, 31, 34syl2anc 397 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
366, 35exlimddv 1794 . . . . . 6 ((𝜑𝑧 ∈ On) → ∃𝐴 𝑧 ∈ dom )
37 eliun 3688 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
3836, 37sylibr 141 . . . . 5 ((𝜑𝑧 ∈ On) → 𝑧 𝐴 dom )
3938ex 112 . . . 4 (𝜑 → (𝑧 ∈ On → 𝑧 𝐴 dom ))
4039ssrdv 2978 . . 3 (𝜑 → On ⊆ 𝐴 dom )
411recsfval 5961 . . . . 5 recs(𝐹) = 𝐴
4241dmeqi 4563 . . . 4 dom recs(𝐹) = dom 𝐴
43 dmuni 4572 . . . 4 dom 𝐴 = 𝐴 dom
4442, 43eqtri 2076 . . 3 dom recs(𝐹) = 𝐴 dom
4540, 44syl6sseqr 3019 . 2 (𝜑 → On ⊆ dom recs(𝐹))
464, 45eqssd 2989 1 (𝜑 → dom recs(𝐹) = On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  Vcvv 2574  cun 2942  wss 2944  {csn 3402  cop 3405   cuni 3607   ciun 3684  Ord word 4126  Oncon0 4127  dom cdm 4372  cres 4374  Fun wfun 4923   Fn wfn 4924  cfv 4929  recscrecs 5949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-recs 5950
This theorem is referenced by:  tfri1d  5979
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