Theorem spwval3 8662

Description: Value of a supremum.

Hypotheses

Ref	Expression
spwval3.1	|- X = U.U.R
spwval3.2	|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))

Assertion

Ref	Expression
spwval3	|- ((R e. U /\ A e. W /\ E.x e. X ph) -> (R supw A) = U.{x e. X | ph})

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

Proof of Theorem spwval3

Step	Hyp	Ref	Expression
1		spwval3.1	. . . 4 |- X = U.U.R
2		spwval3.2	. . . . . 6 |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))
3	2	a1i 8	. . . . 5 |- (x e. X -> (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy))))
4	3	rabbii 1812	. . . 4 |- {x e. X | ph} = {x e. X | (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy))}
5	1, 4	spwval2 8661	. . 3 |- ((R e. U /\ A e. W) -> (R supw A) = if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X))
6	5	3adant3 803	. 2 |- ((R e. U /\ A e. W /\ E.x e. X ph) -> (R supw A) = if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X))
7		rabn0 2302	. . . 4 |- ({x e. X | ph} =/= (/) <-> E.x e. X ph)
8		iftrue 2376	. . . 4 |- ({x e. X | ph} =/= (/) -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X) = U.{x e. X | ph})
9	7, 8	sylbir 201	. . 3 |- (E.x e. X ph -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X) = U.{x e. X | ph})
10	9	3ad2ant3 806	. 2 |- ((R e. U /\ A e. W /\ E.x e. X ph) -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X) = U.{x e. X | ph})
11	6, 10	eqtrd 1514	1 |- ((R e. U /\ A e. W /\ E.x e. X ph) -> (R supw A) = U.{x e. X | ph})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 779   = wceq 960   e. wcel 962   =/= wne 1592  A.wral 1652  E.wrex 1653  {crab 1655  (/)c0 2289  ifcif 2371  P~cpw 2411  U.cuni 2515   class class class wbr 2632  (class class class)co 3977   sup_w cspw 8642

This theorem is referenced by:  spwval 8667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-rab 1659  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-if 2372  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fv 3212  df-opr 3979  df-oprab 3980  df-spw 8648

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