Theorem predidm 6151

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 6129	. 2 ⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋}))
2		df-pred 6129	. . . . 5 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3		inidm 4109	. . . . . 6 ⊢ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})) = (◡𝑅 “ {𝑋})
4	3	ineq2i 4100	. . . . 5 ⊢ (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋}))) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
5	2, 4	eqtr4i 2764	. . . 4 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})))
6		inass 4110	. . . 4 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})))
7	5, 6	eqtr4i 2764	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋}))
8	2	ineq1i 4099	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋}))
9	7, 8	eqtr4i 2764	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋}))
10	1, 9	eqtr4i 2764	1 ⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)

Syntax hints:   = wceq 1542   ∩ cin 3842  {csn 4516  ^◡ccnv 5524   “ cima 5528  Predcpred 6128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-in 3850  df-pred 6129

This theorem is referenced by: (None)