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Theorem bnj1371 32309
Description: Technical lemma for bnj60 32342. 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 32119 . . . . . 6 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3 rexex 3227 . . . . . 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 1848 . . . . 5 (∃𝑦𝜏′ ↔ ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
74, 6sylib 220 . . . 4 (𝑓𝐻 → ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
8 exsimpl 1869 . . . 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))}
1110abeq2i 2946 . . . . . 6 (𝑓𝐶 ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
1211bnj1238 32086 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
13 fnfun 6429 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1412, 13bnj31 31997 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1514bnj1265 32092 . . 3 (𝑓𝐶 → Fun 𝑓)
169, 15bnj593 32024 . 2 (𝑓𝐻 → ∃𝑦Fun 𝑓)
1716bnj937 32051 1 (𝑓𝐻 → Fun 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wex 1780  wcel 2114  {cab 2798  wne 3006  wral 3125  wrex 3126  {crab 3129  [wsbc 3752  cun 3911  wss 3913  c0 4269  {csn 4543  cop 4549   cuni 4814   class class class wbr 5042  dom cdm 5531  cres 5533  Fun wfun 6325   Fn wfn 6326  cfv 6331   predc-bnj14 31966   FrSe w-bnj15 31970   trClc-bnj18 31972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-ral 3130  df-rex 3131  df-fn 6334
This theorem is referenced by:  bnj1384  32312
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