Theorem reuuni2 2884

Description: U.{x e. A | ph} is an explicit representation of "the unique element in A such that ph."

Hypothesis

Ref	Expression
reuuni2.1	|- (x = B -> (ph <-> ps))

Assertion

Ref	Expression
reuuni2	|- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))

Distinct variable groups:   x,A   x,B   ps,x

Proof of Theorem reuuni2

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (y e. B -> A.x y e. B)
2		ax-17 971	. . 3 |- (ps -> A.xps)
3	2	a1i 8	. 2 |- (B e. A -> (ps -> A.xps))
4		reuuni2.1	. 2 |- (x = B -> (ph <-> ps))
5	1, 3, 4	reuuni2f 2883	1 |- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E!wreu 1647  {crab 1648  U.cuni 2503

This theorem is referenced by:  reuuni3 2886  rabsnt 2894  f1ocnvfv3 3883  supub 4580  suplub 4581  suppr 4590  supsnALT 4592  lbinfm 6048  supxr 6081  flval2t 6238  flbit 6240  uzinfm 6462  isumclimtf 7195  grpidinv2 8060  grpinv 8069  spwpr4OLD 8663  spwpr4aOLD 8664  pjeq2t 9241  pjpj0 9255  adjvalvalt 9861  cnlnadjlem5 10004  cnvbravalt 10043  cdj3lem2 10362

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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