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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 5396 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4726 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | releqi 5359 | . 2 ⊢ (Rel ∩ 𝐴 ↔ Rel ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | sylibr 224 | 1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 3051 ∩ cint 4627 ∩ ciin 4673 Rel wrel 5271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-v 3342 df-in 3722 df-ss 3729 df-int 4628 df-iin 4675 df-rel 5273 |
This theorem is referenced by: clrellem 38449 |
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