Metamath Proof Explorer


Theorem trpredeq1

Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011)

Ref Expression
Assertion trpredeq1 R = S TrPred R A X = TrPred S A X

Proof

Step Hyp Ref Expression
1 predeq1 R = S Pred R A y = Pred S A y
2 1 iuneq2d R = S y a Pred R A y = y a Pred S A y
3 2 mpteq2dv R = S a V y a Pred R A y = a V y a Pred S A y
4 predeq1 R = S Pred R A X = Pred S A X
5 rdgeq12 a V y a Pred R A y = a V y a Pred S A y Pred R A X = Pred S A X rec a V y a Pred R A y Pred R A X = rec a V y a Pred S A y Pred S A X
6 3 4 5 syl2anc R = S rec a V y a Pred R A y Pred R A X = rec a V y a Pred S A y Pred S A X
7 6 reseq1d R = S rec a V y a Pred R A y Pred R A X ω = rec a V y a Pred S A y Pred S A X ω
8 7 rneqd R = S ran rec a V y a Pred R A y Pred R A X ω = ran rec a V y a Pred S A y Pred S A X ω
9 8 unieqd R = S ran rec a V y a Pred R A y Pred R A X ω = ran rec a V y a Pred S A y Pred S A X ω
10 df-trpred TrPred R A X = ran rec a V y a Pred R A y Pred R A X ω
11 df-trpred TrPred S A X = ran rec a V y a Pred S A y Pred S A X ω
12 9 10 11 3eqtr4g R = S TrPred R A X = TrPred S A X