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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 5052 | . . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | |
2 | 1 | nfdm 5976 | . . 3 ⊢ Ⅎ𝑥dom ∩ 𝑥 ∈ 𝐴 𝐵 |
3 | 2 | ssiinf 5077 | . 2 ⊢ (dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∀𝑥 ∈ 𝐴 dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵) |
4 | iinss2 5080 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) | |
5 | dmss 5927 | . . 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 3074 | 1 ⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 ⊆ wss 3976 ∩ ciin 5016 dom cdm 5700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-iin 5018 df-br 5167 df-dm 5710 |
This theorem is referenced by: (None) |
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