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Mirrors > Home > ILE Home > Th. List > iinxprg | GIF version |
Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.) |
Ref | Expression |
---|---|
iinxprg.1 | ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) |
iinxprg.2 | ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
iinxprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinxprg.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) | |
2 | 1 | eleq2d 2207 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
3 | iinxprg.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
4 | 3 | eleq2d 2207 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐸)) |
5 | 2, 4 | ralprg 3569 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸))) |
6 | 5 | abbidv 2255 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)}) |
7 | df-iin 3811 | . 2 ⊢ ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} | |
8 | df-in 3072 | . 2 ⊢ (𝐷 ∩ 𝐸) = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)} | |
9 | 6, 7, 8 | 3eqtr4g 2195 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cab 2123 ∀wral 2414 ∩ cin 3065 {cpr 3523 ∩ ciin 3809 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-sn 3528 df-pr 3529 df-iin 3811 |
This theorem is referenced by: (None) |
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