Theorem bnj149 13241

Description: Technical lemma of bnj151 13243. 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
bnj149.1	|- (th1 <-> ((R FrSe A /\ x e. A) -> E*f(f Fn 1o /\ ph' /\ ps')))
bnj149.2	|- (ze0 <-> (f Fn 1o /\ ph' /\ ps'))
bnj149.3	|- (ze1 <-> [g / f]ze0)
bnj149.4	|- (ph1 <-> [g / f]ph')
bnj149.5	|- (ps1 <-> [g / f]ps')
bnj149.6	|- (ph' <-> (f` (/)) = pred(x, A, R))

Assertion

Ref	Expression
bnj149	|- th1

Distinct variable groups:   A,f,g,x   R,f,g,x   f,ze1   g,ze0

Proof of Theorem bnj149

Step	Hyp	Ref	Expression
1		bnj148 12481	. . . . . . . . 9 |- (((/) e. _V /\ pred(x, A, R) e. _V) -> ((f Fn {(/)} /\ (f` (/)) = pred(x, A, R)) <-> f = {<.(/), pred(x, A, R)>.}))
2		0ex 3446	. . . . . . . . 9 |- (/) e. _V
3		bnj93 13217	. . . . . . . . 9 |- ((R FrSe A /\ x e. A) -> pred(x, A, R) e. _V)
4	1, 2, 3	sylancr 526	. . . . . . . 8 |- ((R FrSe A /\ x e. A) -> ((f Fn {(/)} /\ (f` (/)) = pred(x, A, R)) <-> f = {<.(/), pred(x, A, R)>.}))
5		bnj149.2	. . . . . . . . . . . 12 |- (ze0 <-> (f Fn 1o /\ ph' /\ ps'))
6	5	biimpi 168	. . . . . . . . . . 11 |- (ze0 -> (f Fn 1o /\ ph' /\ ps'))
7		df-3an 860	. . . . . . . . . . 11 |- ((f Fn 1o /\ ph' /\ ps') <-> ((f Fn 1o /\ ph') /\ ps'))
8	6, 7	sylib 215	. . . . . . . . . 10 |- (ze0 -> ((f Fn 1o /\ ph') /\ ps'))
9	8	simplld 348	. . . . . . . . 9 |- (ze0 -> (f Fn 1o /\ ph'))
10		df1o2 5185	. . . . . . . . . . 11 |- 1o = {(/)}
11	10	fneq2i 4508	. . . . . . . . . 10 |- (f Fn 1o <-> f Fn {(/)})
12		bnj149.6	. . . . . . . . . 10 |- (ph' <-> (f` (/)) = pred(x, A, R))
13	11, 12	anbi12i 540	. . . . . . . . 9 |- ((f Fn 1o /\ ph') <-> (f Fn {(/)} /\ (f` (/)) = pred(x, A, R)))
14	9, 13	sylib 215	. . . . . . . 8 |- (ze0 -> (f Fn {(/)} /\ (f` (/)) = pred(x, A, R)))
15	4, 14	syl5bi 225	. . . . . . 7 |- ((R FrSe A /\ x e. A) -> (ze0 -> f = {<.(/), pred(x, A, R)>.}))
16		bnj148 12481	. . . . . . . . 9 |- (((/) e. _V /\ pred(x, A, R) e. _V) -> ((g Fn {(/)} /\ (g` (/)) = pred(x, A, R)) <-> g = {<.(/), pred(x, A, R)>.}))
17	16, 2, 3	sylancr 526	. . . . . . . 8 |- ((R FrSe A /\ x e. A) -> ((g Fn {(/)} /\ (g` (/)) = pred(x, A, R)) <-> g = {<.(/), pred(x, A, R)>.}))
18		bnj149.3	. . . . . . . . . . . . 13 |- (ze1 <-> [g / f]ze0)
19		bnj149.4	. . . . . . . . . . . . 13 |- (ph1 <-> [g / f]ph')
20		bnj149.5	. . . . . . . . . . . . 13 |- (ps1 <-> [g / f]ps')
21	5, 18, 19, 20	bnj156 12482	. . . . . . . . . . . 12 |- (ze1 <-> (g Fn 1o /\ ph1 /\ ps1))
22	21	biimpi 168	. . . . . . . . . . 11 |- (ze1 -> (g Fn 1o /\ ph1 /\ ps1))
23		df-3an 860	. . . . . . . . . . 11 |- ((g Fn 1o /\ ph1 /\ ps1) <-> ((g Fn 1o /\ ph1) /\ ps1))
24	22, 23	sylib 215	. . . . . . . . . 10 |- (ze1 -> ((g Fn 1o /\ ph1) /\ ps1))
25	24	simplld 348	. . . . . . . . 9 |- (ze1 -> (g Fn 1o /\ ph1))
26	10	fneq2i 4508	. . . . . . . . . 10 |- (g Fn 1o <-> g Fn {(/)})
27	12	sbbii 1538	. . . . . . . . . . 11 |- ([g / f]ph' <-> [g / f](f` (/)) = pred(x, A, R))
28		visset 2295	. . . . . . . . . . . 12 |- g e. _V
29		fveq1 4680	. . . . . . . . . . . . 13 |- (f = g -> (f` (/)) = (g` (/)))
30	29	eqeq1d 1892	. . . . . . . . . . . 12 |- (f = g -> ((f` (/)) = pred(x, A, R) <-> (g` (/)) = pred(x, A, R)))
31	28, 30	sbcie 2485	. . . . . . . . . . 11 |- ([g / f](f` (/)) = pred(x, A, R) <-> (g` (/)) = pred(x, A, R))
32	19, 27, 31	3bitri 194	. . . . . . . . . 10 |- (ph1 <-> (g` (/)) = pred(x, A, R))
33	26, 32	anbi12i 540	. . . . . . . . 9 |- ((g Fn 1o /\ ph1) <-> (g Fn {(/)} /\ (g` (/)) = pred(x, A, R)))
34	25, 33	sylib 215	. . . . . . . 8 |- (ze1 -> (g Fn {(/)} /\ (g` (/)) = pred(x, A, R)))
35	17, 34	syl5bi 225	. . . . . . 7 |- ((R FrSe A /\ x e. A) -> (ze1 -> g = {<.(/), pred(x, A, R)>.}))
36	15, 35	anim12d 617	. . . . . 6 |- ((R FrSe A /\ x e. A) -> ((ze0 /\ ze1) -> (f = {<.(/), pred(x, A, R)>.} /\ g = {<.(/), pred(x, A, R)>.})))
37		eqtr3 1907	. . . . . 6 |- ((f = {<.(/), pred(x, A, R)>.} /\ g = {<.(/), pred(x, A, R)>.}) -> f = g)
38	36, 37	syl6 25	. . . . 5 |- ((R FrSe A /\ x e. A) -> ((ze0 /\ ze1) -> f = g))
39	38	19.21aivv 1665	. . . 4 |- ((R FrSe A /\ x e. A) -> A.fA.g((ze0 /\ ze1) -> f = g))
40		sbequ12 1545	. . . . . 6 |- (f = g -> ((f Fn 1o /\ ph' /\ ps') <-> [g / f](f Fn 1o /\ ph' /\ ps')))
41	5	sbbii 1538	. . . . . . 7 |- ([g / f]ze0 <-> [g / f](f Fn 1o /\ ph' /\ ps'))
42	18, 41	bitri 190	. . . . . 6 |- (ze1 <-> [g / f](f Fn 1o /\ ph' /\ ps'))
43	40, 5, 42	3bitr4g 614	. . . . 5 |- (f = g -> (ze0 <-> ze1))
44	43	mo4 1799	. . . 4 |- (E*fze0 <-> A.fA.g((ze0 /\ ze1) -> f = g))
45	39, 44	sylibr 217	. . 3 |- ((R FrSe A /\ x e. A) -> E*fze0)
46	5	mobii 1801	. . 3 |- (E*fze0 <-> E*f(f Fn 1o /\ ph' /\ ps'))
47	45, 46	sylib 215	. 2 |- ((R FrSe A /\ x e. A) -> E*f(f Fn 1o /\ ph' /\ ps'))
48		bnj149.1	. 2 |- (th1 <-> ((R FrSe A /\ x e. A) -> E*f(f Fn 1o /\ ph' /\ ps')))
49	47, 48	mpbir 207	1 |- th1

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  E*wmo 1772  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046   Fn wfn 3993  ` cfv 3998  1oc1o 5172   predsyn-bnj14 12023   FrSe syn-bnj15 12027

This theorem is referenced by:  bnj151 13243

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-13 1311  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  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-reu 2111  df-v 2294  df-sbc 2454  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-suc 3663  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-bnj13 12026  df-bnj15 12028

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