Theorem predidm 5702

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 5680	. 2 ⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋}))
2		df-pred 5680	. . . . 5 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3		inidm 3822	. . . . . 6 ⊢ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})) = (◡𝑅 “ {𝑋})
4	3	ineq2i 3811	. . . . 5 ⊢ (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋}))) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
5	2, 4	eqtr4i 2647	. . . 4 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})))
6		inass 3823	. . . 4 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ ((◡𝑅 “ {𝑋}) ∩ (◡𝑅 “ {𝑋})))
7	5, 6	eqtr4i 2647	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋}))
8	2	ineq1i 3810	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (◡𝑅 “ {𝑋}))
9	7, 8	eqtr4i 2647	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (◡𝑅 “ {𝑋}))
10	1, 9	eqtr4i 2647	1 ⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)

Syntax hints:   = wceq 1483   ∩ cin 3573  {csn 4177  ^◡ccnv 5113   “ cima 5117  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-pred 5680

This theorem is referenced by: (None)