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Mirrors > Home > MPE Home > Th. List > predin | Structured version Visualization version GIF version |
Description: Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.) |
Ref | Expression |
---|---|
predin | ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inindir 3793 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋}))) | |
2 | df-pred 5597 | . 2 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) | |
3 | df-pred 5597 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
4 | df-pred 5597 | . . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋})) | |
5 | 3, 4 | ineq12i 3774 | . 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋}))) |
6 | 1, 2, 5 | 3eqtr4i 2642 | 1 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∩ cin 3539 {csn 4125 ◡ccnv 5037 “ cima 5041 Predcpred 5596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-pred 5597 |
This theorem is referenced by: (None) |
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