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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 4912 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 5054 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4545 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | dfdm4 4726 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 4, 5 | xpeq12i 4556 | . 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴) |
7 | 3, 6 | sseqtrri 3127 | 1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3066 × cxp 4532 ◡ccnv 4533 dom cdm 4534 ran crn 4535 Rel wrel 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: (None) |
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