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Mirrors > Home > MPE Home > Th. List > reliin | Structured version Visualization version GIF version |
Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
reliin | ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinss 4507 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | |
2 | df-rel 5045 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
3 | 2 | rexbii 3023 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) |
4 | df-rel 5045 | . 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | |
5 | 1, 3, 4 | 3imtr4i 280 | 1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∩ ciin 4456 × cxp 5036 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-iin 4458 df-rel 5045 |
This theorem is referenced by: relint 5165 xpiindi 5179 dibglbN 35473 dihglbcpreN 35607 |
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