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Theorem bnj1374 35024
Description: Technical lemma for bnj60 35055. 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
bnj1374.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1374.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1374.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1374.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1374.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1374.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1374.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1374.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1374.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
Assertion
Ref Expression
bnj1374 (𝑓𝐻𝑓𝐶)
Distinct variable groups:   𝑥,𝐴   𝐵,𝑓   𝑦,𝐶   𝑥,𝑅   𝑓,𝑑,𝑥   𝑦,𝑓,𝑥
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑦,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦,𝑓,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1374
StepHypRef Expression
1 bnj1374.9 . . . . . 6 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
21bnj1436 34832 . . . . 5 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3 rexex 3074 . . . . 5 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ → ∃𝑦𝜏′)
42, 3syl 17 . . . 4 (𝑓𝐻 → ∃𝑦𝜏′)
5 bnj1374.1 . . . . . 6 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
6 bnj1374.2 . . . . . 6 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7 bnj1374.3 . . . . . 6 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
8 bnj1374.4 . . . . . 6 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
9 bnj1374.8 . . . . . 6 (𝜏′[𝑦 / 𝑥]𝜏)
105, 6, 7, 8, 9bnj1373 35023 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
1110exbii 1845 . . . 4 (∃𝑦𝜏′ ↔ ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
124, 11sylib 218 . . 3 (𝑓𝐻 → ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
13 exsimpl 1866 . . 3 (∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 𝑓𝐶)
1412, 13syl 17 . 2 (𝑓𝐻 → ∃𝑦 𝑓𝐶)
1514bnj937 34764 1 (𝑓𝐻𝑓𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  [wsbc 3791  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637   class class class wbr 5148  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563   predc-bnj14 34681   FrSe w-bnj15 34685   trClc-bnj18 34687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-br 5149  df-bnj14 34682  df-bnj18 34688
This theorem is referenced by:  bnj1384  35025
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