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| Mirrors > Home > MPE Home > Th. List > dprdf2 | Structured version Visualization version GIF version | ||
| Description: The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) |
| Ref | Expression |
|---|---|
| dprdcntz.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
| dprdcntz.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
| Ref | Expression |
|---|---|
| dprdf2 | ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dprdcntz.1 | . . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
| 2 | dprdf 20026 | . . 3 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) |
| 4 | dprdcntz.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
| 5 | 4 | feq2d 6722 | . 2 ⊢ (𝜑 → (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺))) |
| 6 | 3, 5 | mpbid 232 | 1 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 SubGrpcsubg 19138 DProd cdprd 20013 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-ixp 8938 df-dprd 20015 |
| This theorem is referenced by: dprdff 20032 dprdfid 20037 dprdfinv 20039 dprdfadd 20040 dprdfeq0 20042 dprdres 20048 dprdss 20049 dprdf1o 20052 dprdf1 20053 subgdprd 20055 dmdprdsplitlem 20057 dprdcntz2 20058 dpjlem 20071 dpjcntz 20072 dpjdisj 20073 dpjlsm 20074 dpjf 20077 dpjidcl 20078 dpjlid 20081 dpjghm 20083 dpjghm2 20084 ablfac1c 20091 ablfac1eulem 20092 ablfac1eu 20093 ablfaclem2 20106 ablfaclem3 20107 dchrptlem3 27310 |
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