Theorem zorn2lem5 4775

Description: Lemma for zorn2 4779.

Hypotheses

Ref	Expression
zorn2lem.1	|- A e. V
zorn2lem.2	|- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}
zorn2lem.3	|- F = U.B
zorn2lem.4	|- C = {z e. A | A.g e. ran f gRz}
zorn2lem.5	|- D = {z e. A | A.g e. (F"x)gRz}
zorn2lem.6	|- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}
zorn2lem.7	|- H = {z e. A | A.g e. (F"y)gRz}

Assertion

Ref	Expression
zorn2lem5	|- (((w We A /\ x e. On) /\ A.y e. x H =/= (/)) -> (F"x) (_ A)

Distinct variable groups:   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   x,H,u,v,f,t

Proof of Theorem zorn2lem5

Step	Hyp	Ref	Expression
1		ax-17 970	. . . . 5 |- ((w We A /\ x e. On) -> A.y(w We A /\ x e. On))
2		hbra1 1685	. . . . 5 |- (A.y e. x H =/= (/) -> A.yA.y e. x H =/= (/))
3	1, 2	hban 1008	. . . 4 |- (((w We A /\ x e. On) /\ A.y e. x H =/= (/)) -> A.y((w We A /\ x e. On) /\ A.y e. x H =/= (/)))
4		ax-17 970	. . . 4 |- (s e. A -> A.y s e. A)
5		eleq1 1532	. . . . . . . . . . . . . 14 |- ((F` y) = s -> ((F` y) e. A <-> s e. A))
6		zorn2lem.1	. . . . . . . . . . . . . . . 16 |- A e. V
7		zorn2lem.2	. . . . . . . . . . . . . . . 16 |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}
8		zorn2lem.3	. . . . . . . . . . . . . . . 16 |- F = U.B
9		zorn2lem.4	. . . . . . . . . . . . . . . 16 |- C = {z e. A | A.g e. ran f gRz}
10		zorn2lem.7	. . . . . . . . . . . . . . . 16 |- H = {z e. A | A.g e. (F"y)gRz}
11		zorn2lem.6	. . . . . . . . . . . . . . . 16 |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}
12	6, 7, 8, 9, 10, 11	zorn2lem1 4771	. . . . . . . . . . . . . . 15 |- ((y e. On /\ (w We A /\ H =/= (/))) -> (F` y) e. H)
13		ssrab2 2128	. . . . . . . . . . . . . . . . 17 |- {z e. A | A.g e. (F"y)gRz} (_ A
14	10, 13	eqsstr 2088	. . . . . . . . . . . . . . . 16 |- H (_ A
15	14	sseli 2062	. . . . . . . . . . . . . . 15 |- ((F` y) e. H -> (F` y) e. A)
16	12, 15	syl 10	. . . . . . . . . . . . . 14 |- ((y e. On /\ (w We A /\ H =/= (/))) -> (F` y) e. A)
17	5, 16	syl5bi 208	. . . . . . . . . . . . 13 |- ((F` y) = s -> ((y e. On /\ (w We A /\ H =/= (/))) -> s e. A))
18		onelon 2968	. . . . . . . . . . . . 13 |- ((x e. On /\ y e. x) -> y e. On)
19	17, 18	sylani 464	. . . . . . . . . . . 12 |- ((F` y) = s -> (((x e. On /\ y e. x) /\ (w We A /\ H =/= (/))) -> s e. A))
20	19	com12 11	. . . . . . . . . . 11 |- (((x e. On /\ y e. x) /\ (w We A /\ H =/= (/))) -> ((F` y) = s -> s e. A))
21	20	exp43 384	. . . . . . . . . 10 |- (x e. On -> (y e. x -> (w We A -> (H =/= (/) -> ((F` y) = s -> s e. A)))))
22	21	com3r 35	. . . . . . . . 9 |- (w We A -> (x e. On -> (y e. x -> (H =/= (/) -> ((F` y) = s -> s e. A)))))
23	22	imp 350	. . . . . . . 8 |- ((w We A /\ x e. On) -> (y e. x -> (H =/= (/) -> ((F` y) = s -> s e. A))))
24	23	a2d 13	. . . . . . 7 |- ((w We A /\ x e. On) -> ((y e. x -> H =/= (/)) -> (y e. x -> ((F` y) = s -> s e. A))))
25	24	a4sd 984	. . . . . 6 |- ((w We A /\ x e. On) -> (A.y(y e. x -> H =/= (/)) -> (y e. x -> ((F` y) = s -> s e. A))))
26		df-ral 1647	. . . . . 6 |- (A.y e. x H =/= (/) <-> A.y(y e. x -> H =/= (/)))
27	25, 26	syl5ib 206	. . . . 5 |- ((w We A /\ x e. On) -> (A.y e. x H =/= (/) -> (y e. x -> ((F` y) = s -> s e. A))))
28	27	imp 350	. . . 4 |- (((w We A /\ x e. On) /\ A.y e. x H =/= (/)) -> (y e. x -> ((F` y) = s -> s e. A)))
29	3, 4, 28	r19.23ad 1743	. . 3 |- (((w We A /\ x e. On) /\ A.y e. x H =/= (/)) -> (E.y e. x (F` y) = s -> s e. A))
30	7, 8	tfrlem7 3912	. . . 4 |- Fun F
31		fvelima 3759	. . . 4 |- ((Fun F /\ s e. (F"x)) -> E.y e. x (F` y) = s)
32	30, 31	mpan 694	. . 3 |- (s e. (F"x) -> E.y e. x (F` y) = s)
33	29, 32	syl5 21	. 2 |- (((w We A /\ x e. On) /\ A.y e. x H =/= (/)) -> (s e. (F"x) -> s e. A))
34	33	ssrdv 2067	1 |- (((w We A /\ x e. On) /\ A.y e. x H =/= (/)) -> (F"x) (_ A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  {cab 1462   =/= wne 1583  A.wral 1643  E.wrex 1644  {crab 1646  Vcvv 1808   (_ wss 2044  (/)c0 2277  U.cuni 2499   class class class wbr 2615  {copab 2662   We wwe 2912  Oncon0 2944  ran crn 3167   |` cres 3168  "cima 3169  Fun wfun 3172   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  zorn2lem6 4776  zorn2lem7 4777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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