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Theorem gaset 19312
Description: The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
gaset ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)

Proof of Theorem gaset
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2736 . . . 4 (+g𝐺) = (+g𝐺)
3 eqid 2736 . . . 4 (0g𝐺) = (0g𝐺)
41, 2, 3isga 19310 . . 3 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simplbi 497 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
65simprd 495 1 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479   × cxp 5682  wf 6556  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  0gc0g 17485  Grpcgrp 18952   GrpAct cga 19308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-ga 19309
This theorem is referenced by:  gass  19320  gasubg  19321  galactghm  19423
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