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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 2107 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
3 | iinxprg.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
4 | 3 | eleq2d 2107 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐸)) |
5 | 2, 4 | ralprg 3421 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸))) |
6 | 5 | abbidv 2155 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)}) |
7 | df-iin 3660 | . 2 ⊢ ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} | |
8 | df-in 2924 | . 2 ⊢ (𝐷 ∩ 𝐸) = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)} | |
9 | 6, 7, 8 | 3eqtr4g 2097 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∩ cin 2916 {cpr 3376 ∩ ciin 3658 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-sn 3381 df-pr 3382 df-iin 3660 |
This theorem is referenced by: (None) |
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