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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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