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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 a subclass of the 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 5323 | . 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 5768 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5528 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | dmxpss 5782 | . . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵 | |
5 | 3, 4 | eqsstri 3831 | . 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵 |
6 | 1, 5 | eqsstri 3831 | 1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3769 × cxp 5310 ◡ccnv 5311 dom cdm 5312 ran crn 5313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-dm 5322 df-rn 5323 |
This theorem is referenced by: ssxpb 5785 ssrnres 5789 funssxp 6276 fconst 6306 dff2 6597 dff3 6598 fliftf 6793 marypha1lem 8581 marypha1 8582 dfac12lem2 9254 brdom4 9640 nqerf 10040 xptrrel 14062 lern 17540 cnconst2 21416 lmss 21431 tsmsxplem1 22284 causs 23424 i1f0 23795 itg10 23796 taylf 24456 perpln2 25962 locfinref 30424 sitg0 30924 noextendseq 32333 heicant 33933 rntrclfvOAI 38040 rtrclex 38707 trclexi 38710 rtrclexi 38711 cnvtrcl0 38716 rntrcl 38718 brtrclfv2 38802 rp-imass 38847 xphe 38857 rfovcnvf1od 39080 |
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