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Theorem tfrlemi14d 6334
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 6319 . . 3 Ord dom recs(𝐹)
3 ordsson 4492 . . 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 6333 . . . . . . 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 5306 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
11 raleq 2673 . . . . . . . . . . . . 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 2842 . . . . . . . . . . 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 2741 . . . . . . . . . . 11 𝑔 ∈ V
161, 15tfrlem3a 6311 . . . . . . . . . 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 6326 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
19 vex 2741 . . . . . . . . . . . 12 𝑧 ∈ V
205tfrlem3-2d 6313 . . . . . . . . . . . . 13 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
2120simprd 114 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑔) ∈ V)
22 opexg 4229 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
2319, 21, 22sylancr 414 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
24 snidg 3622 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
25 elun2 3304 . . . . . . . . . . 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 4833 . . . . . . . . . . 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 4828 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3332eleq2d 2247 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3433rspcev 2842 . . . . . . . 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 1898 . . . . . 6 ((𝜑𝑧 ∈ On) → ∃𝐴 𝑧 ∈ dom )
37 eliun 3891 . . . . . 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 3162 . . 3 (𝜑 → On ⊆ 𝐴 dom )
411recsfval 6316 . . . . 5 recs(𝐹) = 𝐴
4241dmeqi 4829 . . . 4 dom recs(𝐹) = dom 𝐴
43 dmuni 4838 . . . 4 dom 𝐴 = 𝐴 dom
4442, 43eqtri 2198 . . 3 dom recs(𝐹) = 𝐴 dom
4540, 44sseqtrrdi 3205 . 2 (𝜑 → On ⊆ dom recs(𝐹))
464, 45eqssd 3173 1 (𝜑 → dom recs(𝐹) = On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2738  cun 3128  wss 3130  {csn 3593  cop 3596   cuni 3810   ciun 3887  Ord word 4363  Oncon0 4364  dom cdm 4627  cres 4629  Fun wfun 5211   Fn wfn 5212  cfv 5217  recscrecs 6305
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306
This theorem is referenced by:  tfri1d  6336
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