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Mirrors > Home > ILE Home > Th. List > dmiin | GIF version |
Description: Domain of an intersection. (Contributed by FL, 15-Oct-2012.) |
Ref | Expression |
---|---|
dmiin | ⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfii1 3844 | . . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | |
2 | 1 | nfdm 4783 | . . 3 ⊢ Ⅎ𝑥dom ∩ 𝑥 ∈ 𝐴 𝐵 |
3 | 2 | ssiinf 3862 | . 2 ⊢ (dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∀𝑥 ∈ 𝐴 dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) |
4 | iinss2 3865 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) | |
5 | dmss 4738 | . . 3 ⊢ (∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 → dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝑥 ∈ 𝐴 → dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) |
7 | 3, 6 | mprgbir 2490 | 1 ⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ⊆ wss 3071 ∩ ciin 3814 dom cdm 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-iin 3816 df-br 3930 df-dm 4549 |
This theorem is referenced by: (None) |
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