Theorem predidm 6164

Description: Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)

Assertion

Ref	Expression
predidm	⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)

Proof of Theorem predidm

Step	Hyp	Ref	Expression
1		df-pred 6142	. 2 ⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋}))
2		df-pred 6142	. . . . 5 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3		inidm 4194	. . . . . 6 ⊢ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})) = (◡𝑅 “ {𝑋})
4	3	ineq2i 4185	. . . . 5 ⊢ (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋}))) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
5	2, 4	eqtr4i 2847	. . . 4 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})))
6		inass 4195	. . . 4 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})))
7	5, 6	eqtr4i 2847	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋}))
8	2	ineq1i 4184	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋}))
9	7, 8	eqtr4i 2847	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋}))
10	1, 9	eqtr4i 2847	1 ⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)

Syntax hints:   = wceq 1533   ∩ cin 3934  {csn 4560  ^◡ccnv 5548   “ cima 5552  Predcpred 6141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942  df-pred 6142

This theorem is referenced by: (None)