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Theorem bnj1245 33683
Description: Technical lemma for bnj60 33731. 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 33477 . . 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 33682 . . 3 𝐶 = 𝐾
82, 7eleqtrdi 2844 . 2 (𝜑𝐾)
96eqabi 2878 . . . . . 6 (𝐾 ↔ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍)))
109bnj1238 33475 . . . . 5 (𝐾 → ∃𝑑𝐵 Fn 𝑑)
1110bnj1196 33463 . . . 4 (𝐾 → ∃𝑑(𝑑𝐵 Fn 𝑑))
12 bnj1245.1 . . . . . . 7 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
1312eqabi 2878 . . . . . 6 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1413simplbi 499 . . . . 5 (𝑑𝐵𝑑𝐴)
15 fndm 6606 . . . . 5 ( Fn 𝑑 → dom = 𝑑)
1614, 15bnj1241 33476 . . . 4 ((𝑑𝐵 Fn 𝑑) → dom 𝐴)
1711, 16bnj593 33414 . . 3 (𝐾 → ∃𝑑dom 𝐴)
1817bnj937 33440 . 2 (𝐾 → dom 𝐴)
198, 18syl 17 1 (𝜑 → dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wne 2940  wral 3061  wrex 3070  {crab 3406  cin 3910  wss 3911  cop 4593   class class class wbr 5106  dom cdm 5634  cres 5636   Fn wfn 6492  cfv 6497  w-bnj17 33355   predc-bnj14 33357   FrSe w-bnj15 33361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-bnj17 33356
This theorem is referenced by: (None)
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