Theorem predun 6349

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 4286	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∪ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6321	. 2 ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = ((𝐴 ∪ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6321	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6321	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	uneq12i 4166	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∪ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2775	1 ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1540   ∪ cun 3949   ∩ cin 3950  {csn 4626  ^◡ccnv 5684   “ cima 5688  Predcpred 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-pred 6321

This theorem is referenced by: (None)