Theorem bnj65 13202

Description: Technical lemma of bnj60 13563. 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).

Hypotheses

Ref	Expression
bnj65.1	|- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
bnj65.2	|- Y = <.x, (f |` pred(x, A, R))>.
bnj65.3	|- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
bnj65.4	|- Z = <.x, (g |` pred(x, A, R))>.

Assertion

Ref	Expression
bnj65	|- (g e. C -> Rel g)

Distinct variable groups:   B,f,g   f,G,g   g,Y   f,Z   f,d,g   x,f,g

Proof of Theorem bnj65

Step	Hyp	Ref	Expression
1		bnj65.1	. . . 4 |- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
2		bnj65.2	. . . 4 |- Y = <.x, (f |` pred(x, A, R))>.
3		bnj65.3	. . . 4 |- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
4		bnj65.4	. . . 4 |- Z = <.x, (g |` pred(x, A, R))>.
5	1, 2, 3, 4	bnj64 13201	. . 3 |- C = {g | E.d e. B (g Fn d /\ A.x e. d (g` x) = (G` Z))}
6	5	bnj1436 13130	. 2 |- (g e. C -> E.d e. B (g Fn d /\ A.x e. d (g` x) = (G` Z)))
7		bnj1239 13007	. 2 |- (E.d e. B (g Fn d /\ A.x e. d (g` x) = (G` Z)) -> E.d e. B g Fn d)
8		rexex 2154	. . 3 |- (E.d e. B g Fn d -> E.d g Fn d)
9		ax-17 1317	. . . 4 |- (Rel g -> A.dRel g)
10		fnrel 4511	. . . 4 |- (g Fn d -> Rel g)
11	9, 10	19.23ai 1412	. . 3 |- (E.d g Fn d -> Rel g)
12	8, 11	syl 12	. 2 |- (E.d e. B g Fn d -> Rel g)
13	6, 7, 12	3syl 24	1 |- (g e. C -> Rel g)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  <.cop 3046   |` cres 3988  Rel wrel 3991   Fn wfn 3993  ` cfv 3998   predsyn-bnj14 12023

This theorem is referenced by:  bnj66 13203

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-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-fn 4009  df-fv 4014

Copyright terms: Public domain