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Theorem tfrlemibxssdm 6190
Description: The union of 𝐵 is defined on all ordinals. Lemma for tfrlemi1 6195. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
tfrlemi1.3 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
tfrlemi1.4 (𝜑𝑥 ∈ On)
tfrlemi1.5 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
Assertion
Ref Expression
tfrlemibxssdm (𝜑𝑥 ⊆ dom 𝐵)
Distinct variable groups:   𝑓,𝑔,,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑤,𝐵,𝑓,𝑔,,𝑧   𝜑,𝑔,,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemibxssdm
StepHypRef Expression
1 tfrlemi1.5 . . 3 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2 tfrlemi1.4 . . . 4 (𝜑𝑥 ∈ On)
3 tfrlemisucfn.2 . . . . . . . . . . . 12 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
43tfrlem3-2d 6175 . . . . . . . . . . 11 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 113 . . . . . . . . . 10 (𝜑 → (𝐹𝑔) ∈ V)
653ad2ant1 985 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝐹𝑔) ∈ V)
7 vex 2661 . . . . . . . . . . . . 13 𝑧 ∈ V
8 opexg 4118 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
97, 5, 8sylancr 408 . . . . . . . . . . . 12 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
10 snidg 3522 . . . . . . . . . . . 12 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
11 elun2 3212 . . . . . . . . . . . 12 (⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩} → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
129, 10, 113syl 17 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
13123ad2ant1 985 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
14 simp2r 991 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑧𝑥)
15 simp3l 992 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔 Fn 𝑧)
16 onelon 4274 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
17 rspe 2456 . . . . . . . . . . . . . . 15 ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
1816, 17sylan 279 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
19 tfrlemisucfn.1 . . . . . . . . . . . . . . 15 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
20 vex 2661 . . . . . . . . . . . . . . 15 𝑔 ∈ V
2119, 20tfrlem3a 6173 . . . . . . . . . . . . . 14 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2218, 21sylibr 133 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔𝐴)
23223adant1 982 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔𝐴)
2414, 15, 233jca 1144 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝑧𝑥𝑔 Fn 𝑧𝑔𝐴))
25 snexg 4076 . . . . . . . . . . . . . 14 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
26 unexg 4332 . . . . . . . . . . . . . . 15 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
2720, 26mpan 418 . . . . . . . . . . . . . 14 ({⟨𝑧, (𝐹𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
289, 25, 273syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
29 isset 2664 . . . . . . . . . . . . 13 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V ↔ ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3028, 29sylib 121 . . . . . . . . . . . 12 (𝜑 → ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
31303ad2ant1 985 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
32 simpr3 972 . . . . . . . . . . . . . . 15 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
33 19.8a 1552 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
34 rspe 2456 . . . . . . . . . . . . . . . . 17 ((𝑧𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
35 tfrlemi1.3 . . . . . . . . . . . . . . . . . 18 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
3635abeq2i 2226 . . . . . . . . . . . . . . . . 17 (𝐵 ↔ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3734, 36sylibr 133 . . . . . . . . . . . . . . . 16 ((𝑧𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐵)
3833, 37sylan2 282 . . . . . . . . . . . . . . 15 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐵)
3932, 38eqeltrrd 2193 . . . . . . . . . . . . . 14 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵)
40393exp2 1186 . . . . . . . . . . . . 13 (𝑧𝑥 → (𝑔 Fn 𝑧 → (𝑔𝐴 → ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))))
41403imp 1158 . . . . . . . . . . . 12 ((𝑧𝑥𝑔 Fn 𝑧𝑔𝐴) → ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))
4241exlimdv 1773 . . . . . . . . . . 11 ((𝑧𝑥𝑔 Fn 𝑧𝑔𝐴) → (∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))
4324, 31, 42sylc 62 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵)
44 elunii 3709 . . . . . . . . . 10 ((⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵)
4513, 43, 44syl2anc 406 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵)
46 opeq2 3674 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐹𝑔)⟩)
4746eleq1d 2184 . . . . . . . . . . 11 (𝑤 = (𝐹𝑔) → (⟨𝑧, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵))
4847spcegv 2746 . . . . . . . . . 10 ((𝐹𝑔) ∈ V → (⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵 → ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵))
497eldm2 4705 . . . . . . . . . 10 (𝑧 ∈ dom 𝐵 ↔ ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵)
5048, 49syl6ibr 161 . . . . . . . . 9 ((𝐹𝑔) ∈ V → (⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵𝑧 ∈ dom 𝐵))
516, 45, 50sylc 62 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑧 ∈ dom 𝐵)
52513expia 1166 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥)) → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5352exlimdv 1773 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥)) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5453anassrs 395 . . . . 5 (((𝜑𝑥 ∈ On) ∧ 𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5554ralimdva 2474 . . . 4 ((𝜑𝑥 ∈ On) → (∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑧𝑥 𝑧 ∈ dom 𝐵))
562, 55mpdan 415 . . 3 (𝜑 → (∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑧𝑥 𝑧 ∈ dom 𝐵))
571, 56mpd 13 . 2 (𝜑 → ∀𝑧𝑥 𝑧 ∈ dom 𝐵)
58 dfss3 3055 . 2 (𝑥 ⊆ dom 𝐵 ↔ ∀𝑧𝑥 𝑧 ∈ dom 𝐵)
5957, 58sylibr 133 1 (𝜑𝑥 ⊆ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945  wal 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2391  wrex 2392  Vcvv 2658  cun 3037  wss 3039  {csn 3495  cop 3498   cuni 3704  Oncon0 4253  dom cdm 4507  cres 4509  Fun wfun 5085   Fn wfn 5086  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-tr 3995  df-iord 4256  df-on 4258  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  tfrlemibfn  6191
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