Theorem zornlem3 3605

Description: Lemma for Zorn's lemma.

Hypotheses

Ref	Expression
zornlem.1	|- A e. V
zornlem.2	|- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}
zornlem.3	|- F = U.B
zornlem.4	|- C = {z e. A | A.g e. ran fgRz}
zornlem.5	|- D = {z e. A | A.g e. (F"x)gRz}
zornlem.6	|- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}

Assertion

Ref	Expression
zornlem3	|- ((R Po A /\ (x e. On /\ (w We A /\ -. D = (/)))) -> (y e. x -> -. (F` x) = (F` y)))

Distinct variable group(s):   x,y,w,h,t,z,f,g,u,v,A   B,h,t,f   x,F,y,z,v,u,f,g,h,t   h,G,t,f   t,C   y,D,u,v,f,t   x,R,y,z,w,g,u,v,f,t

Proof of Theorem zornlem3

Step	Hyp	Ref	Expression
1		zornlem.1	. . . 4 |- A e. V
2		zornlem.2	. . . 4 |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}
3		zornlem.3	. . . 4 |- F = U.B
4		zornlem.4	. . . 4 |- C = {z e. A | A.g e. ran fgRz}
5		zornlem.5	. . . 4 |- D = {z e. A | A.g e. (F"x)gRz}
6		zornlem.6	. . . 4 |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}
7	1, 2, 3, 4, 5, 6	zornlem2 3604	. . 3 |- ((x e. On /\ (w We A /\ -. D = (/))) -> (y e. x -> (F` y)R(F` x)))
8	7	adantl 305	. 2 |- ((R Po A /\ (x e. On /\ (w We A /\ -. D = (/)))) -> (y e. x -> (F` y)R(F` x)))
9		breq1 2065	. . . . . 6 |- ((F` x) = (F` y) -> ((F` x)R(F` x) <-> (F` y)R(F` x)))
10	9	biimprcd 138	. . . . 5 |- ((F` y)R(F` x) -> ((F` x) = (F` y) -> (F` x)R(F` x)))
11		poirr 2133	. . . . 5 |- ((R Po A /\ (F` x) e. A) -> -. (F` x)R(F` x))
12	10, 11	nsyli 106	. . . 4 |- ((F` y)R(F` x) -> ((R Po A /\ (F` x) e. A) -> -. (F` x) = (F` y)))
13	12	com12 13	. . 3 |- ((R Po A /\ (F` x) e. A) -> ((F` y)R(F` x) -> -. (F` x) = (F` y)))
14	1, 2, 3, 4, 5, 6	zornlem1 3603	. . . 4 |- ((x e. On /\ (w We A /\ -. D = (/))) -> (F` x) e. D)
15		ssrab 1556	. . . . . 6 |- {z e. A | A.g e. (F"x)gRz} (_ A
16	5, 15	eqsstr 1530	. . . . 5 |- D (_ A
17	16	sseli 1504	. . . 4 |- ((F` x) e. D -> (F` x) e. A)
18	14, 17	syl 12	. . 3 |- ((x e. On /\ (w We A /\ -. D = (/))) -> (F` x) e. A)
19	13, 18	sylan2 346	. 2 |- ((R Po A /\ (x e. On /\ (w We A /\ -. D = (/)))) -> ((F` y)R(F` x) -> -. (F` x) = (F` y)))
20	8, 19	syld 27	1 |- ((R Po A /\ (x e. On /\ (w We A /\ -. D = (/)))) -> (y e. x -> -. (F` x) = (F` y)))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  {crab 1204  Vcvv 1348  (/)c0 1707  U.cuni 1919   class class class wbr 2054  {copab 2055   Po wpo 2058   We wwe 2062  Oncon0 2199  ran crn 2411   |` cres 2412  "cima 2413   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  zornlem4 3606

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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