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Mirrors > Home > ILE Home > Th. List > xpriindim | GIF version |
Description: Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.) |
Ref | Expression |
---|---|
xpriindim | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpindi 4764 | . 2 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) | |
2 | xpiindim 4766 | . . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) | |
3 | 2 | ineq2d 3338 | . 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))) |
4 | 1, 3 | eqtrid 2222 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∩ cin 3130 ∩ ciin 3889 × cxp 4626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-iin 3891 df-opab 4067 df-xp 4634 df-rel 4635 |
This theorem is referenced by: (None) |
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