![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnvssrndm | Structured version Visualization version GIF version |
Description: The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
cnvssrndm | ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5934 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 6088 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 5530 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | dfdm4 5728 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 4, 5 | xpeq12i 5547 | . 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴) |
7 | 3, 6 | sseqtrri 3952 | 1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3881 × cxp 5517 ◡ccnv 5518 dom cdm 5519 ran crn 5520 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-pr 5295 |
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-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 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: wuncnv 10141 fcnvgreu 30436 trclubgNEW 40318 |
Copyright terms: Public domain | W3C validator |