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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 4603 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | |
2 | df-rel 5150 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
3 | 2 | rexbii 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) |
4 | df-rel 5150 | . 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | |
5 | 1, 3, 4 | 3imtr4i 281 | 1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 2942 Vcvv 3231 ⊆ wss 3607 ∩ ciin 4553 × cxp 5141 Rel wrel 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-v 3233 df-in 3614 df-ss 3621 df-iin 4555 df-rel 5150 |
This theorem is referenced by: relint 5275 xpiindi 5290 dibglbN 36772 dihglbcpreN 36906 |
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