![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rnxpss | Structured version Visualization version GIF version |
Description: The range of a Cartesian product is included in its second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
rnxpss | ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5699 | . 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 6178 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5917 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | dmxpss 6192 | . . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵 | |
5 | 3, 4 | eqsstri 4029 | . 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵 |
6 | 1, 5 | eqsstri 4029 | 1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3962 × cxp 5686 ◡ccnv 5687 dom cdm 5688 ran crn 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 |
This theorem is referenced by: ssxpb 6195 ssrnres 6199 resssxp 6291 funssxp 6764 fconst 6794 dff2 7118 dff3 7119 fliftf 7334 frxp2 8167 frxp3 8174 marypha1lem 9470 marypha1 9471 dfac12lem2 10182 brdom4 10567 nqerf 10967 xptrrel 15015 lern 18648 cnconst2 23306 lmss 23321 tsmsxplem1 24176 causs 25345 i1f0 25735 itg10 25736 taylf 26416 noextendseq 27726 perpln2 28733 gsumpart 33042 locfinref 33801 sitg0 34327 heicant 37641 rntrclfvOAI 42678 rtrclex 43606 trclexi 43609 rtrclexi 43610 cnvtrcl0 43615 rntrcl 43617 brtrclfv2 43716 xphe 43770 rfovcnvf1od 43993 |
Copyright terms: Public domain | W3C validator |