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Mirrors > Home > MPE Home > Th. List > reldmghm | Structured version Visualization version GIF version |
Description: Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
reldmghm | ⊢ Rel dom GrpHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ghm 18355 | . 2 ⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∣ [(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑔‘(𝑥(+g‘𝑠)𝑦)) = ((𝑔‘𝑥)(+g‘𝑡)(𝑔‘𝑦)))}) | |
2 | 1 | reldmmpo 7284 | 1 ⊢ Rel dom GrpHom |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 {cab 2799 ∀wral 3138 [wsbc 3771 dom cdm 5554 Rel wrel 5559 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Grpcgrp 18102 GrpHom cghm 18354 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-dm 5564 df-oprab 7159 df-mpo 7160 df-ghm 18355 |
This theorem is referenced by: (None) |
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