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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 4154 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋}))) | |
2 | df-pred 6116 | . 2 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) | |
3 | df-pred 6116 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
4 | df-pred 6116 | . . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋})) | |
5 | 3, 4 | ineq12i 4137 | . 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋}))) |
6 | 1, 2, 5 | 3eqtr4i 2831 | 1 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∩ cin 3880 {csn 4525 ◡ccnv 5518 “ cima 5522 Predcpred 6115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-pred 6116 |
This theorem is referenced by: (None) |
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