Theorem mouniss 2896

Description: Restriction of a unique element to a smaller class.

Assertion

Ref	Expression
mouniss	|- ((A (_ B /\ E.x e. A ph /\ E*x(x e. B /\ ph)) -> U.{x e. A | ph} = U.{x e. B | ph})

Distinct variable groups:   x,A   x,B

Proof of Theorem mouniss

Step	Hyp	Ref	Expression
1		ssel 2066	. . . . . . . 8 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 562	. . . . . . 7 |- (A (_ B -> ((x e. A /\ ph) -> (x e. B /\ ph)))
3	2	r19.22dv2 1739	. . . . . 6 |- (A (_ B -> (E.x e. A ph -> E.x e. B ph))
4	3	imp 350	. . . . 5 |- ((A (_ B /\ E.x e. A ph) -> E.x e. B ph)
5	4	anim1i 334	. . . 4 |- (((A (_ B /\ E.x e. A ph) /\ E*x(x e. B /\ ph)) -> (E.x e. B ph /\ E*x(x e. B /\ ph)))
6		reu5 1932	. . . 4 |- (E!x e. B ph <-> (E.x e. B ph /\ E*x(x e. B /\ ph)))
7	5, 6	sylibr 200	. . 3 |- (((A (_ B /\ E.x e. A ph) /\ E*x(x e. B /\ ph)) -> E!x e. B ph)
8		reuuniss 2895	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. A | ph} = U.{x e. B | ph})
9	8	3expa 835	. . 3 |- (((A (_ B /\ E.x e. A ph) /\ E!x e. B ph) -> U.{x e. A | ph} = U.{x e. B | ph})
10	7, 9	syldan 469	. 2 |- (((A (_ B /\ E.x e. A ph) /\ E*x(x e. B /\ ph)) -> U.{x e. A | ph} = U.{x e. B | ph})
11	10	3impa 830	1 |- ((A (_ B /\ E.x e. A ph /\ E*x(x e. B /\ ph)) -> U.{x e. A | ph} = U.{x e. B | ph})

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E*wmo 1383  E.wrex 1649  E!wreu 1650  {crab 1651   (_ wss 2050  U.cuni 2507

This theorem is referenced by:  adjbdlnt 10011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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