Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1371 Structured version   Visualization version   GIF version

Theorem bnj1371 31434
Description: Technical lemma for bnj60 31467. 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 31247 . . . . . 6 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3 rexex 3150 . . . . . 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 1924 . . . . 5 (∃𝑦𝜏′ ↔ ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
74, 6sylib 208 . . . 4 (𝑓𝐻 → ∃𝑦(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
8 exsimpl 1946 . . . 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 2884 . . . . . 6 (𝑓𝐶 ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
1211bnj1238 31214 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
13 fnfun 6127 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1412, 13bnj31 31124 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1514bnj1265 31220 . . 3 (𝑓𝐶 → Fun 𝑓)
169, 15bnj593 31152 . 2 (𝑓𝐻 → ∃𝑦Fun 𝑓)
1716bnj937 31179 1 (𝑓𝐻 → Fun 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  [wsbc 3587  cun 3721  wss 3723  c0 4063  {csn 4317  cop 4323   cuni 4575   class class class wbr 4787  dom cdm 5250  cres 5252  Fun wfun 6024   Fn wfn 6025  cfv 6030   predc-bnj14 31093   FrSe w-bnj15 31097   trClc-bnj18 31099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-rex 3067  df-fn 6033
This theorem is referenced by:  bnj1384  31437
  Copyright terms: Public domain W3C validator