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Mirrors > Home > ILE Home > Th. List > riinm | GIF version |
Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.) |
Ref | Expression |
---|---|
riinm | ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3339 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) | |
2 | r19.2m 3521 | . . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) | |
3 | 2 | ancoms 268 | . . . 4 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) |
4 | iinss 3950 | . . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) |
6 | df-ss 3154 | . . 3 ⊢ (∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ↔ (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆) | |
7 | 5, 6 | sylib 122 | . 2 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆) |
8 | 1, 7 | eqtrid 2232 | 1 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∃wex 1502 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 ∩ cin 3140 ⊆ wss 3141 ∩ ciin 3899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-in 3147 df-ss 3154 df-iin 3901 |
This theorem is referenced by: riinerm 6622 |
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