Theorem bnj102 13222

Description: Technical lemma of bnj103 13223. 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).

Hypothesis

Ref	Expression
bnj102.1	|- F = {<.(/), pred(x, A, R)>.}

Assertion

Ref	Expression
bnj102	|- ((R FrSe A /\ x e. A) -> Fun F)

Distinct variable groups:   x,A   x,R

Proof of Theorem bnj102

Step	Hyp	Ref	Expression
1		funsng 4465	. . 3 |- (((/) e. _V /\ pred(x, A, R) e. _V) -> Fun {<.(/), pred(x, A, R)>.})
2		bnj102.1	. . . 4 |- F = {<.(/), pred(x, A, R)>.}
3	2	funeqi 4442	. . 3 |- (Fun F <-> Fun {<.(/), pred(x, A, R)>.})
4	1, 3	sylibr 217	. 2 |- (((/) e. _V /\ pred(x, A, R) e. _V) -> Fun F)
5		0ex 3446	. 2 |- (/) e. _V
6		bnj93 13217	. 2 |- ((R FrSe A /\ x e. A) -> pred(x, A, R) e. _V)
7	4, 5, 6	sylancr 526	1 |- ((R FrSe A /\ x e. A) -> Fun F)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046  Fun wfun 3992   predsyn-bnj14 12023   FrSe syn-bnj15 12027

This theorem is referenced by:  bnj103 13223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-fun 4008  df-bnj13 12026  df-bnj15 12028

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