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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 5240 | . 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4574 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | releqi 5202 | . 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 2913 ∩ cint 4475 ∩ ciin 4521 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-int 4476 df-iin 4523 df-rel 5121 |
This theorem is referenced by: clrellem 37929 |
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