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Mirrors > Home > ILE Home > Th. List > cnvssrndm | 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 5001 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 5144 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4633 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | dfdm4 4814 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 4, 5 | xpeq12i 4644 | . 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴) |
7 | 3, 6 | sseqtrri 3190 | 1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3129 × cxp 4620 ◡ccnv 4621 dom cdm 4622 ran crn 4623 Rel wrel 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4628 df-rel 4629 df-cnv 4630 df-dm 4632 df-rn 4633 |
This theorem is referenced by: (None) |
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