![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > relbrcnvg | Structured version Visualization version GIF version |
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5885 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
relbrcnvg | ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6109 | . . . 4 ⊢ Rel ◡𝑅 | |
2 | 1 | brrelex12i 5733 | . . 3 ⊢ (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | 2 | a1i 11 | . 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
4 | brrelex12 5730 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
5 | 4 | ancomd 460 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
6 | 5 | ex 411 | . 2 ⊢ (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
7 | brcnvg 5882 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | |
8 | 7 | a1i 11 | . 2 ⊢ (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))) |
9 | 3, 6, 8 | pm5.21ndd 378 | 1 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 Vcvv 3461 class class class wbr 5149 ◡ccnv 5677 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 |
This theorem is referenced by: eliniseg2 6111 relbrcnv 6112 isinv 17746 releleccnv 37859 relcnveq2 37925 elrelscnveq2 38095 eqvrelsym 38207 brco2f1o 43604 brco3f1o 43605 ntrclsnvobr 43624 neicvgel1 43691 |
Copyright terms: Public domain | W3C validator |