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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 3709 | . . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | |
2 | 1 | nfdm 4596 | . . 3 ⊢ Ⅎ𝑥dom ∩ 𝑥 ∈ 𝐴 𝐵 |
3 | 2 | ssiinf 3727 | . 2 ⊢ (dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∀𝑥 ∈ 𝐴 dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) |
4 | iinss2 3730 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) | |
5 | dmss 4552 | . . 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 2421 | 1 ⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 ⊆ wss 2973 ∩ ciin 3679 dom cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-iin 3681 df-br 3786 df-dm 4373 |
This theorem is referenced by: (None) |
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