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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 19121 | . . 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 6473 | . 2 ⊢ (𝜑 → (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺))) |
6 | 3, 5 | mpbid 235 | 1 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 SubGrpcsubg 18265 DProd cdprd 19108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-ixp 8445 df-dprd 19110 |
This theorem is referenced by: dprdff 19127 dprdfid 19132 dprdfinv 19134 dprdfadd 19135 dprdfeq0 19137 dprdres 19143 dprdss 19144 dprdf1o 19147 dprdf1 19148 subgdprd 19150 dmdprdsplitlem 19152 dprdcntz2 19153 dpjlem 19166 dpjcntz 19167 dpjdisj 19168 dpjlsm 19169 dpjf 19172 dpjidcl 19173 dpjlid 19176 dpjghm 19178 dpjghm2 19179 ablfac1c 19186 ablfac1eulem 19187 ablfac1eu 19188 ablfaclem2 19201 ablfaclem3 19202 dchrptlem3 25850 |
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