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Mirrors > Home > MPE Home > Th. List > relint | Structured version Visualization version GIF version |
Description: The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
relint | ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reliin 5687 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4968 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | releqi 5649 | . 2 ⊢ (Rel ∩ 𝐴 ↔ Rel ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | sylibr 237 | 1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 3062 ∩ cint 4859 ∩ ciin 4905 Rel wrel 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-v 3410 df-in 3873 df-ss 3883 df-int 4860 df-iin 4907 df-rel 5558 |
This theorem is referenced by: clrellem 40909 |
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