Theorem bnj97 13220

Description: Technical lemma of bnj109 13226. 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
bnj97.1	|- F = {<.(/), pred(x, A, R)>.}

Assertion

Ref	Expression
bnj97	|- ((R FrSe A /\ x e. A) -> (F` (/)) = pred(x, A, R))

Distinct variable groups:   x,A   x,R

Proof of Theorem bnj97

Step	Hyp	Ref	Expression
1		bnj93 13217	. . 3 |- ((R FrSe A /\ x e. A) -> pred(x, A, R) e. _V)
2		0ex 3446	. . . . 5 |- (/) e. _V
3	2	bnj519 12520	. . . 4 |- ( pred(x, A, R) e. _V -> Fun {<.(/), pred(x, A, R)>.})
4		bnj97.1	. . . . 5 |- F = {<.(/), pred(x, A, R)>.}
5	4	funeqi 4442	. . . 4 |- (Fun F <-> Fun {<.(/), pred(x, A, R)>.})
6	3, 5	sylibr 217	. . 3 |- ( pred(x, A, R) e. _V -> Fun F)
7	1, 6	syl 12	. 2 |- ((R FrSe A /\ x e. A) -> Fun F)
8		opex 3527	. . . . 5 |- <.(/), pred(x, A, R)>. e. _V
9	8	snid 3069	. . . 4 |- <.(/), pred(x, A, R)>. e. {<.(/), pred(x, A, R)>.}
10	9, 4	eleqtrri 1970	. . 3 |- <.(/), pred(x, A, R)>. e. F
11		funopfvg 4711	. . . 4 |- (( pred(x, A, R) e. _V /\ Fun F) -> (<.(/), pred(x, A, R)>. e. F -> (F` (/)) = pred(x, A, R)))
12	11, 1	sylan 497	. . 3 |- (((R FrSe A /\ x e. A) /\ Fun F) -> (<.(/), pred(x, A, R)>. e. F -> (F` (/)) = pred(x, A, R)))
13	10, 12	mpi 55	. 2 |- (((R FrSe A /\ x e. A) /\ Fun F) -> (F` (/)) = pred(x, A, R))
14	7, 13	mpdan 768	1 |- ((R FrSe A /\ x e. A) -> (F` (/)) = pred(x, A, R))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046  Fun wfun 3992  ` cfv 3998   predsyn-bnj14 12023   FrSe syn-bnj15 12027

This theorem is referenced by:  bnj109 13226  bnj127 13237

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-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-bnj13 12026  df-bnj15 12028

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