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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 5689 | . 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 6161 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5907 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | dmxpss 6175 | . . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵 | |
5 | 3, 4 | eqsstri 4014 | . 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵 |
6 | 1, 5 | eqsstri 4014 | 1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3947 × cxp 5676 ◡ccnv 5677 dom cdm 5678 ran crn 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: ssxpb 6178 ssrnres 6182 resssxp 6274 funssxp 6752 fconst 6783 dff2 7109 dff3 7110 fliftf 7323 frxp2 8149 frxp3 8156 marypha1lem 9457 marypha1 9458 dfac12lem2 10168 brdom4 10554 nqerf 10954 xptrrel 14960 lern 18583 cnconst2 23200 lmss 23215 tsmsxplem1 24070 causs 25239 i1f0 25629 itg10 25630 taylf 26308 noextendseq 27613 perpln2 28528 gsumpart 32782 locfinref 33442 sitg0 33966 heicant 37128 rntrclfvOAI 42111 rtrclex 43047 trclexi 43050 rtrclexi 43051 cnvtrcl0 43056 rntrcl 43058 brtrclfv2 43157 xphe 43211 rfovcnvf1od 43434 |
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