![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7862 | . 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
2 | dfrel2 6142 | . . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
3 | cnvexg 7862 | . . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V) | |
4 | eleq1 2822 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V)) | |
5 | 3, 4 | imbitrid 243 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
6 | 2, 5 | sylbi 216 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
7 | 1, 6 | impbid2 225 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ◡ccnv 5633 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: f1oexrnex 7865 |
Copyright terms: Public domain | W3C validator |