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Theorem gaset 18083
 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 2825 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2825 . . . 4 (+g𝐺) = (+g𝐺)
3 eqid 2825 . . . 4 (0g𝐺) = (0g𝐺)
41, 2, 3isga 18081 . . 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 493 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
65simprd 491 1 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   × cxp 5344  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  +gcplusg 16312  0gc0g 16460  Grpcgrp 17783   GrpAct cga 18079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-ga 18080 This theorem is referenced by:  gass  18091  gasubg  18092  galactghm  18180
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