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| 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 5056 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | |
| 2 | df-rel 5692 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
| 3 | 2 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | 
| 4 | df-rel 5692 | . 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V)) | |
| 5 | 1, 3, 4 | 3imtr4i 292 | 1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∩ ciin 4992 × cxp 5683 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-iin 4994 df-rel 5692 | 
| This theorem is referenced by: relint 5829 xpiindi 5846 dibglbN 41168 dihglbcpreN 41302 | 
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