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Theorem bnj1371 34887
Description: Technical lemma for bnj60 34920. 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
bnj1371.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1371.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1371.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1371.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1371.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1371.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1371.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1371.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1371.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1371.10 𝑃 = 𝐻
bnj1371.11 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
Assertion
Ref Expression
bnj1371 (𝑓𝐻 → Fun 𝑓)
Distinct variable groups:   𝑓,𝑑   𝑦,𝑓
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑥,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 bnj1371
StepHypRef Expression
1 bnj1371.9 . . . . . . 7 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
21bnj1436 34697 . . . . . 6 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3 rexex 3066 . . . . . 6 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ → ∃𝑦𝜏′)
42, 3syl 17 . . . . 5 (𝑓𝐻 → ∃𝑦𝜏′)
5 bnj1371.11 . . . . . 6 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
65exbii 1843 . . . . 5 (∃𝑦𝜏′ ↔ ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
74, 6sylib 217 . . . 4 (𝑓𝐻 → ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
8 exsimpl 1864 . . . 4 (∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 𝑓𝐶)
97, 8syl 17 . . 3 (𝑓𝐻 → ∃𝑦 𝑓𝐶)
10 bnj1371.3 . . . . . . 7 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
1110eqabri 2870 . . . . . 6 (𝑓𝐶 ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
1211bnj1238 34664 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
13 fnfun 6652 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1412, 13bnj31 34577 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1514bnj1265 34670 . . 3 (𝑓𝐶 → Fun 𝑓)
169, 15bnj593 34603 . 2 (𝑓𝐻 → ∃𝑦Fun 𝑓)
1716bnj937 34629 1 (𝑓𝐻 → Fun 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  {crab 3419  [wsbc 3775  cun 3944  wss 3946  c0 4322  {csn 4623  cop 4629   cuni 4905   class class class wbr 5145  dom cdm 5674  cres 5676  Fun wfun 6540   Fn wfn 6541  cfv 6546   predc-bnj14 34546   FrSe w-bnj15 34550   trClc-bnj18 34552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-fn 6549
This theorem is referenced by:  bnj1384  34890
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