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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 7611 | . 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
2 | dfrel2 6013 | . . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
3 | cnvexg 7611 | . . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V) | |
4 | eleq1 2877 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V)) | |
5 | 3, 4 | syl5ib 247 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
6 | 2, 5 | sylbi 220 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
7 | 1, 6 | impbid2 229 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ◡ccnv 5518 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 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: f1oexrnex 7614 |
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