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Theorem bnj1245 31410
Description: Technical lemma for bnj60 31458. 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 31207 . . 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 31409 . . 3 𝐶 = 𝐾
82, 7syl6eleq 2849 . 2 (𝜑𝐾)
96abeq2i 2873 . . . . . 6 (𝐾 ↔ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍)))
109bnj1238 31205 . . . . 5 (𝐾 → ∃𝑑𝐵 Fn 𝑑)
1110bnj1196 31193 . . . 4 (𝐾 → ∃𝑑(𝑑𝐵 Fn 𝑑))
12 bnj1245.1 . . . . . . 7 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
1312abeq2i 2873 . . . . . 6 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1413simplbi 478 . . . . 5 (𝑑𝐵𝑑𝐴)
15 fndm 6151 . . . . 5 ( Fn 𝑑 → dom = 𝑑)
1614, 15bnj1241 31206 . . . 4 ((𝑑𝐵 Fn 𝑑) → dom 𝐴)
1711, 16bnj593 31143 . . 3 (𝐾 → ∃𝑑dom 𝐴)
1817bnj937 31170 . 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 383  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  {crab 3054  cin 3714  wss 3715  cop 4327   class class class wbr 4804  dom cdm 5266  cres 5268   Fn wfn 6044  cfv 6049  w-bnj17 31082   predc-bnj14 31084   FrSe w-bnj15 31088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-bnj17 31083
This theorem is referenced by: (None)
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