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Mirrors > Home > MPE Home > Th. List > predun | Structured version Visualization version GIF version |
Description: Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.) |
Ref | Expression |
---|---|
predun | ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indir 4222 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∪ (𝐵 ∩ (◡𝑅 “ {𝑋}))) | |
2 | df-pred 6238 | . 2 ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = ((𝐴 ∪ 𝐵) ∩ (◡𝑅 “ {𝑋})) | |
3 | df-pred 6238 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
4 | df-pred 6238 | . . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋})) | |
5 | 3, 4 | uneq12i 4108 | . 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∪ (𝐵 ∩ (◡𝑅 “ {𝑋}))) |
6 | 1, 2, 5 | 3eqtr4i 2774 | 1 ⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∪ cun 3896 ∩ cin 3897 {csn 4573 ◡ccnv 5619 “ cima 5623 Predcpred 6237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3404 df-v 3443 df-un 3903 df-in 3905 df-pred 6238 |
This theorem is referenced by: (None) |
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