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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 5648 | . 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 6113 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5864 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | dmxpss 6127 | . . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵 | |
5 | 3, 4 | eqsstri 3982 | . 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵 |
6 | 1, 5 | eqsstri 3982 | 1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3914 × cxp 5635 ◡ccnv 5636 dom cdm 5637 ran crn 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 |
This theorem is referenced by: ssxpb 6130 ssrnres 6134 resssxp 6226 funssxp 6701 fconst 6732 dff2 7053 dff3 7054 fliftf 7264 frxp2 8080 frxp3 8087 marypha1lem 9377 marypha1 9378 dfac12lem2 10088 brdom4 10474 nqerf 10874 xptrrel 14874 lern 18488 cnconst2 22657 lmss 22672 tsmsxplem1 23527 causs 24685 i1f0 25074 itg10 25075 taylf 25743 noextendseq 27038 perpln2 27702 gsumpart 31953 locfinref 32486 sitg0 33010 heicant 36163 rntrclfvOAI 41061 rtrclex 41981 trclexi 41984 rtrclexi 41985 cnvtrcl0 41990 rntrcl 41992 brtrclfv2 42091 xphe 42145 rfovcnvf1od 42368 |
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