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Theorem bnj1245 32396
Description: Technical lemma for bnj60 32444. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1245.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1245.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1245.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1245.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1245.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1245.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1245.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
bnj1245.8 𝑍 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1245.9 𝐾 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍))}
Assertion
Ref Expression
bnj1245 (𝜑 → dom 𝐴)
Distinct variable groups:   𝐴,𝑑   𝐵,𝑓,   𝑓,𝐺,   ,𝑌   𝑓,𝑍   𝑓,𝑑,   𝑥,𝑓,
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑓,𝑔,)   𝐵(𝑥,𝑦,𝑔,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑅(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑔,𝑑)   𝐾(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑔,𝑑)   𝑍(𝑥,𝑦,𝑔,,𝑑)

Proof of Theorem bnj1245
StepHypRef Expression
1 bnj1245.6 . . . 4 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
21bnj1247 32190 . . 3 (𝜑𝐶)
3 bnj1245.2 . . . 4 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4 bnj1245.3 . . . 4 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
5 bnj1245.8 . . . 4 𝑍 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
6 bnj1245.9 . . . 4 𝐾 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍))}
73, 4, 5, 6bnj1234 32395 . . 3 𝐶 = 𝐾
82, 7eleqtrdi 2900 . 2 (𝜑𝐾)
96abeq2i 2925 . . . . . 6 (𝐾 ↔ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍)))
109bnj1238 32188 . . . . 5 (𝐾 → ∃𝑑𝐵 Fn 𝑑)
1110bnj1196 32176 . . . 4 (𝐾 → ∃𝑑(𝑑𝐵 Fn 𝑑))
12 bnj1245.1 . . . . . . 7 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
1312abeq2i 2925 . . . . . 6 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1413simplbi 501 . . . . 5 (𝑑𝐵𝑑𝐴)
15 fndm 6425 . . . . 5 ( Fn 𝑑 → dom = 𝑑)
1614, 15bnj1241 32189 . . . 4 ((𝑑𝐵 Fn 𝑑) → dom 𝐴)
1711, 16bnj593 32126 . . 3 (𝐾 → ∃𝑑dom 𝐴)
1817bnj937 32153 . 2 (𝐾 → dom 𝐴)
198, 18syl 17 1 (𝜑 → dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  {cab 2776  wne 2987  wral 3106  wrex 3107  {crab 3110  cin 3880  wss 3881  cop 4531   class class class wbr 5030  dom cdm 5519  cres 5521   Fn wfn 6319  cfv 6324  w-bnj17 32066   predc-bnj14 32068   FrSe w-bnj15 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-bnj17 32067
This theorem is referenced by: (None)
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