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Mirrors > Home > MPE Home > Th. List > relcnvexb | Structured version Visualization version GIF version |
Description: A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Ref | Expression |
---|---|
relcnvexb | ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvexg 7259 | . 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
2 | dfrel2 5724 | . . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
3 | cnvexg 7259 | . . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V) | |
4 | eleq1 2838 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V)) | |
5 | 3, 4 | syl5ib 234 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
6 | 2, 5 | sylbi 207 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
7 | 1, 6 | impbid2 216 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ◡ccnv 5248 Rel wrel 5254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-cnv 5257 df-dm 5259 df-rn 5260 |
This theorem is referenced by: f1oexrnex 7262 |
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