Theorem supub 4567

Description: A supremum is an upper bound.

Hypothesis

Ref	Expression
supmo.1	|- R Or A

Assertion

Ref	Expression
supub	|- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, A, R)RC))

Distinct variable groups:   x,y,z,A   x,R,y,z   x,B,y,z

Proof of Theorem supub

Step	Hyp	Ref	Expression
1		breq2 2620	. . . . 5 |- (w = C -> (sup(B, A, R)Rw <-> sup(B, A, R)RC))
2	1	negbid 610	. . . 4 |- (w = C -> (-. sup(B, A, R)Rw <-> -. sup(B, A, R)RC))
3	2	imbi2d 611	. . 3 |- (w = C -> ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)Rw) <-> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)RC)))
4		breq2 2620	. . . . . 6 |- (y = w -> (sup(B, A, R)Ry <-> sup(B, A, R)Rw))
5	4	negbid 610	. . . . 5 |- (y = w -> (-. sup(B, A, R)Ry <-> -. sup(B, A, R)Rw))
6	5	rcla4v 1871	. . . 4 |- (w e. B -> (A.y e. B -. sup(B, A, R)Ry -> -. sup(B, A, R)Rw))
7		df-sup 4561	. . . . . . 7 |- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
8	7	eqcomi 1478	. . . . . 6 |- U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)
9		breq1 2619	. . . . . . . . . . 11 |- (x = sup(B, A, R) -> (xRy <-> sup(B, A, R)Ry))
10	9	negbid 610	. . . . . . . . . 10 |- (x = sup(B, A, R) -> (-. xRy <-> -. sup(B, A, R)Ry))
11	10	ralbidv 1662	. . . . . . . . 9 |- (x = sup(B, A, R) -> (A.y e. B -. xRy <-> A.y e. B -. sup(B, A, R)Ry))
12		breq2 2620	. . . . . . . . . . 11 |- (x = sup(B, A, R) -> (yRx <-> yRsup(B, A, R)))
13	12	imbi1d 612	. . . . . . . . . 10 |- (x = sup(B, A, R) -> ((yRx -> E.z e. B yRz) <-> (yRsup(B, A, R) -> E.z e. B yRz)))
14	13	ralbidv 1662	. . . . . . . . 9 |- (x = sup(B, A, R) -> (A.y e. A (yRx -> E.z e. B yRz) <-> A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)))
15	11, 14	anbi12d 627	. . . . . . . 8 |- (x = sup(B, A, R) -> ((A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) <-> (A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz))))
16	15	reuuni2 2881	. . . . . . 7 |- ((sup(B, A, R) e. A /\ E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))) -> ((A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)) <-> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)))
17		supmo.1	. . . . . . . 8 |- R Or A
18	17	supcl 4566	. . . . . . 7 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> sup(B, A, R) e. A)
19	17	supeu 4565	. . . . . . 7 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
20	16, 18, 19	sylanc 471	. . . . . 6 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)) <-> U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)))
21	8, 20	mpbiri 194	. . . . 5 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (A.y e. B -. sup(B, A, R)Ry /\ A.y e. A (yRsup(B, A, R) -> E.z e. B yRz)))
22	21	pm3.26d 321	. . . 4 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> A.y e. B -. sup(B, A, R)Ry)
23	6, 22	syl5 21	. . 3 |- (w e. B -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)Rw))
24	3, 23	vtoclga 1850	. 2 |- (C e. B -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> -. sup(B, A, R)RC))
25	24	com12 11	1 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, A, R)RC))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1644  E.wrex 1645  E!wreu 1646  {crab 1647  U.cuni 2500   class class class wbr 2616   Or wor 2836  supcsup 4560

This theorem is referenced by:  supubi 4572  supmax 4576  suprub 6017  supxrun 6046  supxrub 6059  gelsupvalOLD 10411

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-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 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-un 2863

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 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-po 2837  df-so 2847  df-sup 4561

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