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Theorem tfrlemi14d 6080
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 6065 . . 3 Ord dom recs(𝐹)
3 ordsson 4299 . . 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 6079 . . . . . . 7 ((𝜑𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
75ad2antrr 472 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
8 simplr 497 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑧 ∈ On)
9 simprl 498 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑔 Fn 𝑧)
10 fneq2 5089 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
11 raleq 2562 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1210, 11anbi12d 457 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1312rspcev 2722 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1413adantll 460 . . . . . . . . . 10 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
15 vex 2622 . . . . . . . . . . 11 𝑔 ∈ V
161, 15tfrlem3a 6057 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1714, 16sylibr 132 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → 𝑔𝐴)
181, 7, 8, 9, 17tfrlemisucaccv 6072 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
19 vex 2622 . . . . . . . . . . . 12 𝑧 ∈ V
205tfrlem3-2d 6059 . . . . . . . . . . . . 13 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
2120simprd 112 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑔) ∈ V)
22 opexg 4046 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
2319, 21, 22sylancr 405 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
24 snidg 3468 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
25 elun2 3166 . . . . . . . . . . 11 (⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩} → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
2623, 24, 253syl 17 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
2726ad2antrr 472 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
28 opeldmg 4629 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
2919, 21, 28sylancr 405 . . . . . . . . . 10 (𝜑 → (⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3029ad2antrr 472 . . . . . . . . 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 4624 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3332eleq2d 2157 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3433rspcev 2722 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
3518, 31, 34syl2anc 403 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
366, 35exlimddv 1826 . . . . . 6 ((𝜑𝑧 ∈ On) → ∃𝐴 𝑧 ∈ dom )
37 eliun 3729 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
3836, 37sylibr 132 . . . . 5 ((𝜑𝑧 ∈ On) → 𝑧 𝐴 dom )
3938ex 113 . . . 4 (𝜑 → (𝑧 ∈ On → 𝑧 𝐴 dom ))
4039ssrdv 3029 . . 3 (𝜑 → On ⊆ 𝐴 dom )
411recsfval 6062 . . . . 5 recs(𝐹) = 𝐴
4241dmeqi 4625 . . . 4 dom recs(𝐹) = dom 𝐴
43 dmuni 4634 . . . 4 dom 𝐴 = 𝐴 dom
4442, 43eqtri 2108 . . 3 dom recs(𝐹) = 𝐴 dom
4540, 44syl6sseqr 3071 . 2 (𝜑 → On ⊆ dom recs(𝐹))
464, 45eqssd 3040 1 (𝜑 → dom recs(𝐹) = On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  Vcvv 2619  cun 2995  wss 2997  {csn 3441  cop 3444   cuni 3648   ciun 3725  Ord word 4180  Oncon0 4181  dom cdm 4428  cres 4430  Fun wfun 4996   Fn wfn 4997  cfv 5002  recscrecs 6051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-recs 6052
This theorem is referenced by:  tfri1d  6082
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