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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 5883 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
| 2 | dmss 5913 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) |
| 4 | df-rn 5696 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
| 5 | df-rn 5696 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
| 6 | 3, 4, 5 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 ◡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-ext 2708 |
| 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-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-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: rnssi 5951 imass1 6119 imass2 6120 ssxpb 6194 sofld 6207 resssxp 6290 funssxp 6764 dff2 7119 dff3 7120 fliftf 7335 1stcof 8044 2ndcof 8045 frxp 8151 frxp2 8169 frxp3 8176 fodomfi 9350 fodomfiOLD 9370 marypha1lem 9473 marypha1 9474 dfac12lem2 10185 fpwwe2lem12 10682 prdsvallem 17499 prdsval 17500 prdsbas 17502 prdsplusg 17503 prdsmulr 17504 prdsvsca 17505 prdshom 17512 catcfuccl 18163 catcxpccl 18252 odf1o2 19591 dprdres 20048 lmss 23306 txss12 23613 txbasval 23614 fmss 23954 tsmsxplem1 24161 ustimasn 24237 utopbas 24244 metustexhalf 24569 causs 25332 ovoliunlem1 25537 dvcnvrelem1 26056 taylf 26402 subgrprop3 29293 sspba 30746 imadifxp 32614 gsumpart 33060 metideq 33892 sxbrsigalem5 34290 omsmon 34300 carsggect 34320 carsgclctunlem2 34321 heicant 37662 mblfinlem1 37664 symrefref2 38564 dicval 41178 aks6d1c2 42131 rntrclfvOAI 42702 diophrw 42770 dnnumch2 43057 lmhmlnmsplit 43099 hbtlem6 43141 mptrcllem 43626 rntrcl 43641 dfrcl2 43687 relexpss1d 43718 rfovcnvf1od 44017 supcnvlimsup 45755 fourierdlem42 46164 sge0less 46407 isubgredgss 47851 |
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