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Mirrors > Home > ILE Home > Th. List > riinint | GIF version |
Description: Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
riinint | ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 4067 | . . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ V) | |
2 | 1 | expcom 115 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑆 ⊆ 𝑋 → 𝑆 ∈ V)) |
3 | 2 | ralimdv 2500 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋 → ∀𝑘 ∈ 𝐼 𝑆 ∈ V)) |
4 | 3 | imp 123 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∀𝑘 ∈ 𝐼 𝑆 ∈ V) |
5 | dfiin3g 4797 | . . . 4 ⊢ (∀𝑘 ∈ 𝐼 𝑆 ∈ V → ∩ 𝑘 ∈ 𝐼 𝑆 = ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ 𝑘 ∈ 𝐼 𝑆 = ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) |
7 | 6 | ineq2d 3277 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) |
8 | intun 3802 | . . 3 ⊢ ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) | |
9 | intsng 3805 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑋} = 𝑋) | |
10 | 9 | adantr 274 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ {𝑋} = 𝑋) |
11 | 10 | ineq1d 3276 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∩ {𝑋} ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) |
12 | 8, 11 | syl5eq 2184 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) = (𝑋 ∩ ∩ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) |
13 | 7, 12 | eqtr4d 2175 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 Vcvv 2686 ∪ cun 3069 ∩ cin 3070 ⊆ wss 3071 {csn 3527 ∩ cint 3771 ∩ ciin 3814 ↦ cmpt 3989 ran crn 4540 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-int 3772 df-iin 3816 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: (None) |
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