ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlemibxssdm GIF version

Theorem tfrlemibxssdm 6274
Description: The union of 𝐵 is defined on all ordinals. Lemma for tfrlemi1 6279. (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 6259 . . . . . . . . . . 11 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 113 . . . . . . . . . 10 (𝜑 → (𝐹𝑔) ∈ V)
653ad2ant1 1003 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝐹𝑔) ∈ V)
7 vex 2715 . . . . . . . . . . . . 13 𝑧 ∈ V
8 opexg 4188 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
97, 5, 8sylancr 411 . . . . . . . . . . . 12 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
10 snidg 3589 . . . . . . . . . . . 12 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
11 elun2 3275 . . . . . . . . . . . 12 (⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩} → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
129, 10, 113syl 17 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
13123ad2ant1 1003 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
14 simp2r 1009 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑧𝑥)
15 simp3l 1010 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔 Fn 𝑧)
16 onelon 4344 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
17 rspe 2506 . . . . . . . . . . . . . . 15 ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
1816, 17sylan 281 . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . 15 𝑔 ∈ V
2119, 20tfrlem3a 6257 . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔𝐴)
2414, 15, 233jca 1162 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝑧𝑥𝑔 Fn 𝑧𝑔𝐴))
25 snexg 4145 . . . . . . . . . . . . . 14 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
26 unexg 4403 . . . . . . . . . . . . . . 15 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
2720, 26mpan 421 . . . . . . . . . . . . . 14 ({⟨𝑧, (𝐹𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
289, 25, 273syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
29 isset 2718 . . . . . . . . . . . . 13 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V ↔ ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3028, 29sylib 121 . . . . . . . . . . . 12 (𝜑 → ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
31303ad2ant1 1003 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
32 simpr3 990 . . . . . . . . . . . . . . 15 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
33 19.8a 1570 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
34 rspe 2506 . . . . . . . . . . . . . . . . 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 2268 . . . . . . . . . . . . . . . . 17 (𝐵 ↔ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3734, 36sylibr 133 . . . . . . . . . . . . . . . 16 ((𝑧𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐵)
3833, 37sylan2 284 . . . . . . . . . . . . . . 15 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐵)
3932, 38eqeltrrd 2235 . . . . . . . . . . . . . 14 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵)
40393exp2 1207 . . . . . . . . . . . . 13 (𝑧𝑥 → (𝑔 Fn 𝑧 → (𝑔𝐴 → ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))))
41403imp 1176 . . . . . . . . . . . 12 ((𝑧𝑥𝑔 Fn 𝑧𝑔𝐴) → ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))
4241exlimdv 1799 . . . . . . . . . . 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 3777 . . . . . . . . . 10 ((⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵)
4513, 43, 44syl2anc 409 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵)
46 opeq2 3742 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐹𝑔)⟩)
4746eleq1d 2226 . . . . . . . . . . 11 (𝑤 = (𝐹𝑔) → (⟨𝑧, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵))
4847spcegv 2800 . . . . . . . . . 10 ((𝐹𝑔) ∈ V → (⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵 → ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵))
497eldm2 4784 . . . . . . . . . 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 1187 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥)) → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5352exlimdv 1799 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥)) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5453anassrs 398 . . . . 5 (((𝜑𝑥 ∈ On) ∧ 𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5554ralimdva 2524 . . . 4 ((𝜑𝑥 ∈ On) → (∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑧𝑥 𝑧 ∈ dom 𝐵))
562, 55mpdan 418 . . 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 3118 . 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 963  wal 1333   = wceq 1335  wex 1472  wcel 2128  {cab 2143  wral 2435  wrex 2436  Vcvv 2712  cun 3100  wss 3102  {csn 3560  cop 3563   cuni 3772  Oncon0 4323  dom cdm 4586  cres 4588  Fun wfun 5164   Fn wfn 5165  cfv 5170
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-iord 4326  df-on 4328  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178
This theorem is referenced by:  tfrlemibfn  6275
  Copyright terms: Public domain W3C validator