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Mirrors > Home > MPE Home > Th. List > dmiin | Structured version Visualization version 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 4925 | . . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | |
2 | 1 | nfdm 5805 | . . 3 ⊢ Ⅎ𝑥dom ∩ 𝑥 ∈ 𝐴 𝐵 |
3 | 2 | ssiinf 4949 | . 2 ⊢ (dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∀𝑥 ∈ 𝐴 dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) |
4 | iinss2 4952 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) | |
5 | dmss 5756 | . . 3 ⊢ (∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 → dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑥 ∈ 𝐴 → dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) |
7 | 3, 6 | mprgbir 3066 | 1 ⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 ⊆ wss 3853 ∩ ciin 4891 dom cdm 5536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-iin 4893 df-br 5040 df-dm 5546 |
This theorem is referenced by: (None) |
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