Theorem suplub 4561

Description: A supremum is the least upper bound.

Hypothesis

Ref	Expression
supmo.1	|- R Or A

Assertion

Ref	Expression
suplub	|- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ CRsup(B, A, R)) -> E.z e. B CRz))

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

Proof of Theorem suplub

Step	Hyp	Ref	Expression
1		breq1 2617	. . . . . 6 |- (w = C -> (wRsup(B, A, R) <-> CRsup(B, A, R)))
2		breq1 2617	. . . . . . 7 |- (w = C -> (wRz <-> CRz))
3	2	rexbidv 1661	. . . . . 6 |- (w = C -> (E.z e. B wRz <-> E.z e. B CRz))
4	1, 3	imbi12d 625	. . . . 5 |- (w = C -> ((wRsup(B, A, R) -> E.z e. B wRz) <-> (CRsup(B, A, R) -> E.z e. B CRz)))
5	4	imbi2d 611	. . . 4 |- (w = C -> ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (wRsup(B, A, R) -> E.z e. B wRz)) <-> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (CRsup(B, A, R) -> E.z e. B CRz))))
6		breq1 2617	. . . . . . 7 |- (y = w -> (yRsup(B, A, R) <-> wRsup(B, A, R)))
7		breq1 2617	. . . . . . . 8 |- (y = w -> (yRz <-> wRz))
8	7	rexbidv 1661	. . . . . . 7 |- (y = w -> (E.z e. B yRz <-> E.z e. B wRz))
9	6, 8	imbi12d 625	. . . . . 6 |- (y = w -> ((yRsup(B, A, R) -> E.z e. B yRz) <-> (wRsup(B, A, R) -> E.z e. B wRz)))
10	9	rcla4v 1869	. . . . 5 |- (w e. A -> (A.y e. A (yRsup(B, A, R) -> E.z e. B yRz) -> (wRsup(B, A, R) -> E.z e. B wRz)))
11		df-sup 4554	. . . . . . . 8 |- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
12	11	eqcomi 1476	. . . . . . 7 |- U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))} = sup(B, A, R)
13		breq1 2617	. . . . . . . . . . . 12 |- (x = sup(B, A, R) -> (xRy <-> sup(B, A, R)Ry))
14	13	negbid 610	. . . . . . . . . . 11 |- (x = sup(B, A, R) -> (-. xRy <-> -. sup(B, A, R)Ry))
15	14	ralbidv 1660	. . . . . . . . . 10 |- (x = sup(B, A, R) -> (A.y e. B -. xRy <-> A.y e. B -. sup(B, A, R)Ry))
16		breq2 2618	. . . . . . . . . . . 12 |- (x = sup(B, A, R) -> (yRx <-> yRsup(B, A, R)))
17	16	imbi1d 612	. . . . . . . . . . 11 |- (x = sup(B, A, R) -> ((yRx -> E.z e. B yRz) <-> (yRsup(B, A, R) -> E.z e. B yRz)))
18	17	ralbidv 1660	. . . . . . . . . 10 |- (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)))
19	15, 18	anbi12d 627	. . . . . . . . 9 |- (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))))
20	19	reuuni2 2879	. . . . . . . 8 |- ((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)))
21		supmo.1	. . . . . . . . 9 |- R Or A
22	21	supcl 4559	. . . . . . . 8 |- (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)
23	21	supeu 4558	. . . . . . . 8 |- (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)))
24	20, 22, 23	sylanc 471	. . . . . . 7 |- (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)))
25	12, 24	mpbiri 194	. . . . . 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)))
26	25	pm3.27d 325	. . . . 5 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> A.y e. A (yRsup(B, A, R) -> E.z e. B yRz))
27	10, 26	syl5 21	. . . 4 |- (w e. A -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (wRsup(B, A, R) -> E.z e. B wRz)))
28	5, 27	vtoclga 1848	. . 3 |- (C e. A -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (CRsup(B, A, R) -> E.z e. B CRz)))
29	28	com12 11	. 2 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. A -> (CRsup(B, A, R) -> E.z e. B CRz)))
30	29	imp3a 361	1 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ CRsup(B, A, R)) -> E.z e. B CRz))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  E!wreu 1644  {crab 1645  U.cuni 2498   class class class wbr 2614   Or wor 2834  supcsup 4553

This theorem is referenced by:  supnub 4562  suplubi 4566  suprlub 6012  supxrun 6040  supxrunb1 6044  supxrunb2 6045

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-po 2835  df-so 2845  df-sup 4554

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