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Theorem bnj1259 32185
Description: Technical lemma for bnj60 32231. 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
bnj1259.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1259.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1259.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1259.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1259.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1259.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1259.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
Assertion
Ref Expression
bnj1259 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓,   𝑓,𝐺,   𝑅,𝑓   ,𝑌   𝑓,𝑑,   𝑥,𝑓,
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑔,,𝑑)   𝐵(𝑥,𝑦,𝑔,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑅(𝑥,𝑦,𝑔,,𝑑)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑔,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑔,𝑑)

Proof of Theorem bnj1259
StepHypRef Expression
1 bnj1259.6 . 2 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
2 abid 2800 . . . 4 ( ∈ { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))} ↔ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
32bnj1238 31977 . . 3 ( ∈ { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑𝐵 Fn 𝑑)
4 bnj1259.2 . . . 4 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
5 bnj1259.3 . . . 4 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
6 eqid 2818 . . . 4 𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
7 eqid 2818 . . . 4 { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
84, 5, 6, 7bnj1234 32182 . . 3 𝐶 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
93, 8eleq2s 2928 . 2 (𝐶 → ∃𝑑𝐵 Fn 𝑑)
101, 9bnj771 31934 1 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  {crab 3139  cin 3932  wss 3933  cop 4563   class class class wbr 5057  dom cdm 5548  cres 5550   Fn wfn 6343  cfv 6348  w-bnj17 31855   predc-bnj14 31857   FrSe w-bnj15 31861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-bnj17 31856
This theorem is referenced by:  bnj1253  32186  bnj1286  32188  bnj1280  32189
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