Metamath Proof Explorer


Theorem bj-imdiridlem

Description: Lemma for bj-imdirid and bj-iminvid . (Contributed by BJ, 26-May-2024)

Ref Expression
Hypothesis bj-imdiridlem.1 xAyAφx=y
Assertion bj-imdiridlem xy|xAyAφ=I𝒫A

Proof

Step Hyp Ref Expression
1 bj-imdiridlem.1 xAyAφx=y
2 1 biimp3a xAyAφx=y
3 2 3expib xAyAφx=y
4 equcomi x=yy=x
5 4 sseq1d x=yyAxA
6 5 biimparc xAx=yyA
7 simpr xAx=yyAyA
8 1 biimpar xAyAx=yφ
9 8 an32s xAx=yyAφ
10 7 9 jca xAx=yyAyAφ
11 6 10 mpdan xAx=yyAφ
12 11 ex xAx=yyAφ
13 3 12 impbid xAyAφx=y
14 13 pm5.32i xAyAφxAx=y
15 anass xAyAφxAyAφ
16 velpw x𝒫AxA
17 vex yV
18 17 ideq xIyx=y
19 16 18 anbi12i x𝒫AxIyxAx=y
20 14 15 19 3bitr4i xAyAφx𝒫AxIy
21 20 opabbii xy|xAyAφ=xy|x𝒫AxIy
22 dfres2 I𝒫A=xy|x𝒫AxIy
23 21 22 eqtr4i xy|xAyAφ=I𝒫A