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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 5692 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4985 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | releqi 5654 | . 2 ⊢ (Rel ∩ 𝐴 ↔ Rel ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | sylibr 236 | 1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 3141 ∩ cint 4878 ∩ ciin 4922 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-in 3945 df-ss 3954 df-int 4879 df-iin 4924 df-rel 5564 |
This theorem is referenced by: clrellem 39989 |
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