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Theorem tfrlemibxssdm 5941
 Description: The union of 𝐵 is defined on all ordinals. Lemma for tfrlemi1 5946. (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 5928 . . . . . . . . . . 11 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 107 . . . . . . . . . 10 (𝜑 → (𝐹𝑔) ∈ V)
653ad2ant1 925 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝐹𝑔) ∈ V)
7 vex 2560 . . . . . . . . . . . . 13 𝑧 ∈ V
8 opexg 3964 . . . . . . . . . . . . 13 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
97, 5, 8sylancr 393 . . . . . . . . . . . 12 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
10 snidg 3400 . . . . . . . . . . . 12 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → ⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩})
11 elun2 3111 . . . . . . . . . . . 12 (⟨𝑧, (𝐹𝑔)⟩ ∈ {⟨𝑧, (𝐹𝑔)⟩} → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
129, 10, 113syl 17 . . . . . . . . . . 11 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
13123ad2ant1 925 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
14 simp2r 931 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑧𝑥)
15 simp3l 932 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔 Fn 𝑧)
16 onelon 4121 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
17 rspe 2370 . . . . . . . . . . . . . . 15 ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
1816, 17sylan 267 . . . . . . . . . . . . . 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 2560 . . . . . . . . . . . . . . 15 𝑔 ∈ V
2119, 20tfrlem3a 5925 . . . . . . . . . . . . . 14 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
2218, 21sylibr 137 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔𝐴)
23223adant1 922 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑔𝐴)
2414, 15, 233jca 1084 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝑧𝑥𝑔 Fn 𝑧𝑔𝐴))
25 snexg 3936 . . . . . . . . . . . . . 14 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
26 unexg 4178 . . . . . . . . . . . . . . 15 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
2720, 26mpan 400 . . . . . . . . . . . . . 14 ({⟨𝑧, (𝐹𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
289, 25, 273syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
29 isset 2561 . . . . . . . . . . . . 13 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V ↔ ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3028, 29sylib 127 . . . . . . . . . . . 12 (𝜑 → ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
31303ad2ant1 925 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
32 simpr3 912 . . . . . . . . . . . . . . 15 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
33 19.8a 1482 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
34 rspe 2370 . . . . . . . . . . . . . . . . 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 2148 . . . . . . . . . . . . . . . . 17 (𝐵 ↔ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})))
3734, 36sylibr 137 . . . . . . . . . . . . . . . 16 ((𝑧𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐵)
3833, 37sylan2 270 . . . . . . . . . . . . . . 15 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐵)
3932, 38eqeltrrd 2115 . . . . . . . . . . . . . 14 ((𝑧𝑥 ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵)
40393exp2 1122 . . . . . . . . . . . . 13 (𝑧𝑥 → (𝑔 Fn 𝑧 → (𝑔𝐴 → ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))))
41403imp 1098 . . . . . . . . . . . 12 ((𝑧𝑥𝑔 Fn 𝑧𝑔𝐴) → ( = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))
4241exlimdv 1700 . . . . . . . . . . 11 ((𝑧𝑥𝑔 Fn 𝑧𝑔𝐴) → (∃ = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵))
4324, 31, 42sylc 56 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵)
44 elunii 3585 . . . . . . . . . 10 ((⟨𝑧, (𝐹𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵)
4513, 43, 44syl2anc 391 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵)
46 opeq2 3550 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐹𝑔)⟩)
4746eleq1d 2106 . . . . . . . . . . 11 (𝑤 = (𝐹𝑔) → (⟨𝑧, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵))
4847spcegv 2641 . . . . . . . . . 10 ((𝐹𝑔) ∈ V → (⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵 → ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵))
497eldm2 4533 . . . . . . . . . 10 (𝑧 ∈ dom 𝐵 ↔ ∃𝑤𝑧, 𝑤⟩ ∈ 𝐵)
5048, 49syl6ibr 151 . . . . . . . . 9 ((𝐹𝑔) ∈ V → (⟨𝑧, (𝐹𝑔)⟩ ∈ 𝐵𝑧 ∈ dom 𝐵))
516, 45, 50sylc 56 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))) → 𝑧 ∈ dom 𝐵)
52513expia 1106 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥)) → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5352exlimdv 1700 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧𝑥)) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5453anassrs 380 . . . . 5 (((𝜑𝑥 ∈ On) ∧ 𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → 𝑧 ∈ dom 𝐵))
5554ralimdva 2387 . . . 4 ((𝜑𝑥 ∈ On) → (∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → ∀𝑧𝑥 𝑧 ∈ dom 𝐵))
562, 55mpdan 398 . . 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 2935 . 2 (𝑥 ⊆ dom 𝐵 ↔ ∀𝑧𝑥 𝑧 ∈ dom 𝐵)
5957, 58sylibr 137 1 (𝜑𝑥 ⊆ dom 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∪ cun 2915   ⊆ wss 2917  {csn 3375  ⟨cop 3378  ∪ cuni 3580  Oncon0 4100  dom cdm 4345   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  tfrlemibfn  5942
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