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Theorem bnj1245 30843
Description: Technical lemma for bnj60 30891. 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 30640 . . 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 30842 . . 3 𝐶 = 𝐾
82, 7syl6eleq 2708 . 2 (𝜑𝐾)
96abeq2i 2732 . . . . . 6 (𝐾 ↔ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍)))
109bnj1238 30638 . . . . 5 (𝐾 → ∃𝑑𝐵 Fn 𝑑)
1110bnj1196 30626 . . . 4 (𝐾 → ∃𝑑(𝑑𝐵 Fn 𝑑))
12 bnj1245.1 . . . . . . 7 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
1312abeq2i 2732 . . . . . 6 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1413simplbi 476 . . . . 5 (𝑑𝐵𝑑𝐴)
15 fndm 5958 . . . . 5 ( Fn 𝑑 → dom = 𝑑)
1614, 15bnj1241 30639 . . . 4 ((𝑑𝐵 Fn 𝑑) → dom 𝐴)
1711, 16bnj593 30576 . . 3 (𝐾 → ∃𝑑dom 𝐴)
1817bnj937 30603 . 2 (𝐾 → dom 𝐴)
198, 18syl 17 1 (𝜑 → dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2908  wrex 2909  {crab 2912  cin 3559  wss 3560  cop 4161   class class class wbr 4623  dom cdm 5084  cres 5086   Fn wfn 5852  cfv 5857  w-bnj17 30512   predc-bnj14 30514   FrSe w-bnj15 30518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865  df-bnj17 30513
This theorem is referenced by: (None)
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