Theorem predun 6351

Description: Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Assertion

Ref	Expression
predun	⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predun

Step	Hyp	Ref	Expression
1		indir 4292	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∪ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6323	. 2 ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = ((𝐴 ∪ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6323	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6323	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	uneq12i 4176	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∪ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2773	1 ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1537   ∪ cun 3961   ∩ cin 3962  {csn 4631  ^◡ccnv 5688   “ cima 5692  Predcpred 6322

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-pred 6323

This theorem is referenced by: (None)