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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 20041 | . . 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 6723 | . 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 1537 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 SubGrpcsubg 19151 DProd cdprd 20028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-ixp 8937 df-dprd 20030 |
This theorem is referenced by: dprdff 20047 dprdfid 20052 dprdfinv 20054 dprdfadd 20055 dprdfeq0 20057 dprdres 20063 dprdss 20064 dprdf1o 20067 dprdf1 20068 subgdprd 20070 dmdprdsplitlem 20072 dprdcntz2 20073 dpjlem 20086 dpjcntz 20087 dpjdisj 20088 dpjlsm 20089 dpjf 20092 dpjidcl 20093 dpjlid 20096 dpjghm 20098 dpjghm2 20099 ablfac1c 20106 ablfac1eulem 20107 ablfac1eu 20108 ablfaclem2 20121 ablfaclem3 20122 dchrptlem3 27325 |
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