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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliind | Structured version Visualization version GIF version |
Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
eliind.a | ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶) |
eliind.k | ⊢ (𝜑 → 𝐾 ∈ 𝐵) |
eliind.d | ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) |
Ref | Expression |
---|---|
eliind | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliind.d | . 2 ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) | |
2 | eliind.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶) | |
3 | eliin 4926 | . . . 4 ⊢ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
5 | 2, 4 | mpbid 234 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
6 | eliind.k | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐵) | |
7 | 1, 5, 6 | rspcdva 3627 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-iin 4924 |
This theorem is referenced by: iooiinioc 41839 hspdifhsp 42905 smflimlem3 43056 smfsuplem1 43092 smflimsuplem4 43104 |
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