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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 5654 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4946 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | releqi 5616 | . 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 3107 ∩ cint 4838 ∩ ciin 4882 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-in 3888 df-ss 3898 df-int 4839 df-iin 4884 df-rel 5526 |
This theorem is referenced by: clrellem 40322 |
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