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Mirrors > Home > MPE Home > Th. List > rnss | Structured version Visualization version GIF version |
Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
rnss | ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 5216 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | dmss 5245 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) |
4 | df-rn 5049 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | df-rn 5049 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
6 | 3, 4, 5 | 3sstr4g 3609 | 1 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3540 ◡ccnv 5037 dom cdm 5038 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: imass1 5419 imass2 5420 ssxpb 5487 ssrnres 5491 sofld 5500 funssxp 5974 fssres 5983 dff2 6279 dff3 6280 fliftf 6465 1stcof 7087 2ndcof 7088 frxp 7174 smores 7336 fodomfi 8124 marypha1lem 8222 marypha1 8223 dfac12lem2 8849 brdom4 9233 smobeth 9287 fpwwe2lem13 9343 nqerf 9631 prdsval 15938 prdsbas 15940 prdsplusg 15941 prdsmulr 15942 prdsvsca 15943 prdshom 15950 catcoppccl 16581 catcfuccl 16582 catcxpccl 16670 lern 17048 odf1o2 17811 gsumzres 18133 gsumzaddlem 18144 gsumzadd 18145 dprdfadd 18242 dprdres 18250 lmss 20912 txss12 21218 txbasval 21219 txkgen 21265 fmss 21560 tsmsxplem1 21766 ustimasn 21842 utopbas 21849 metustexhalf 22171 causs 22904 ovoliunlem1 23077 dvcnvrelem1 23584 taylf 23919 dvlog 24197 perpln2 25406 sspba 26966 imadifxp 28796 metideq 29264 sxbrsigalem5 29677 omsmon 29687 carsggect 29707 carsgclctunlem2 29708 nodenselem6 31085 fixssrn 31184 heicant 32614 mblfinlem1 32616 dicval 35483 rntrclfvOAI 36272 diophrw 36340 dnnumch2 36633 lmhmlnmsplit 36675 hbtlem6 36718 mptrcllem 36939 cnvrcl0 36951 rntrcl 36954 dfrcl2 36985 relexpss1d 37016 rp-imass 37085 rfovcnvf1od 37318 rnresss 38360 fourierdlem42 39042 sge0less 39285 subgrprop3 40500 |
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