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Theorem bnj1374 35007
Description: Technical lemma for bnj60 35038. 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 34815 . . . . 5 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3 rexex 3082 . . . . 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 35006 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
1110exbii 1846 . . . 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 1867 . . 3 (∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 𝑓𝐶)
1412, 13syl 17 . 2 (𝑓𝐻 → ∃𝑦 𝑓𝐶)
1514bnj937 34747 1 (𝑓𝐻𝑓𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wex 1777  wcel 2108  {cab 2717  wne 2946  wral 3067  wrex 3076  {crab 3443  [wsbc 3804  cun 3974  wss 3976  c0 4352  {csn 4648  cop 4654   class class class wbr 5166  dom cdm 5700  cres 5702   Fn wfn 6568  cfv 6573   predc-bnj14 34664   FrSe w-bnj15 34668   trClc-bnj18 34670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-iun 5017  df-br 5167  df-bnj14 34665  df-bnj18 34671
This theorem is referenced by:  bnj1384  35008
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