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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 5696 | . 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
| 2 | cnvxp 6177 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
| 3 | 2 | dmeqi 5915 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
| 4 | dmxpss 6191 | . . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵 | |
| 5 | 3, 4 | eqsstri 4030 | . 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵 |
| 6 | 1, 5 | eqsstri 4030 | 1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: ssxpb 6194 ssrnres 6198 resssxp 6290 funssxp 6764 fconst 6794 dff2 7119 dff3 7120 fliftf 7335 frxp2 8169 frxp3 8176 marypha1lem 9473 marypha1 9474 dfac12lem2 10185 brdom4 10570 nqerf 10970 xptrrel 15019 lern 18636 cnconst2 23291 lmss 23306 tsmsxplem1 24161 causs 25332 i1f0 25722 itg10 25723 taylf 26402 noextendseq 27712 perpln2 28719 gsumpart 33060 locfinref 33840 sitg0 34348 heicant 37662 rntrclfvOAI 42702 rtrclex 43630 trclexi 43633 rtrclexi 43634 cnvtrcl0 43639 rntrcl 43641 brtrclfv2 43740 xphe 43794 rfovcnvf1od 44017 |
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