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Mirrors > Home > ILE Home > Th. List > iinxsng | GIF version |
Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iinxsng.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iinxsng | ⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 3824 | . 2 ⊢ ∩ 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵} | |
2 | iinxsng.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | eleq2d 2210 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
4 | 3 | ralsng 3571 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
5 | 4 | abbi1dv 2260 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵} = 𝐶) |
6 | 1, 5 | syl5eq 2185 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 {csn 3532 ∩ ciin 3822 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-sbc 2914 df-sn 3538 df-iin 3824 |
This theorem is referenced by: (None) |
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