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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 5967 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 6121 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 5566 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | dfdm4 5764 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 4, 5 | xpeq12i 5583 | . 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴) |
7 | 3, 6 | sseqtrri 4004 | 1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3936 × cxp 5553 ◡ccnv 5554 dom cdm 5555 ran crn 5556 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: wuncnv 10152 fcnvgreu 30418 trclubgNEW 39998 |
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