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Theorem gapm 18096
Description: The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
gapm.1 𝑋 = (Base‘𝐺)
gapm.2 𝐹 = (𝑥𝑌 ↦ (𝐴 𝑥))
Assertion
Ref Expression
gapm (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥,   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem gapm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 gapm.2 . 2 𝐹 = (𝑥𝑌 ↦ (𝐴 𝑥))
2 gapm.1 . . . . 5 𝑋 = (Base‘𝐺)
32gaf 18085 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
43ad2antrr 717 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → :(𝑋 × 𝑌)⟶𝑌)
5 simplr 785 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝐴𝑋)
6 simpr 479 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝑥𝑌)
74, 5, 6fovrnd 7071 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → (𝐴 𝑥) ∈ 𝑌)
83ad2antrr 717 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → :(𝑋 × 𝑌)⟶𝑌)
9 gagrp 18082 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 717 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐺 ∈ Grp)
11 simplr 785 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐴𝑋)
12 eqid 2825 . . . . 5 (invg𝐺) = (invg𝐺)
132, 12grpinvcl 17828 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
1410, 11, 13syl2anc 579 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr 479 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
168, 14, 15fovrnd 7071 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → (((invg𝐺)‘𝐴) 𝑦) ∈ 𝑌)
17 simpll 783 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ∈ (𝐺 GrpAct 𝑌))
18 simplr 785 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝐴𝑋)
19 simprl 787 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
20 simprr 789 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
212, 12gacan 18095 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2217, 18, 19, 20, 21syl13anc 1495 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2322bicomd 215 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((((invg𝐺)‘𝐴) 𝑦) = 𝑥 ↔ (𝐴 𝑥) = 𝑦))
24 eqcom 2832 . . 3 (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥)
25 eqcom 2832 . . 3 (𝑦 = (𝐴 𝑥) ↔ (𝐴 𝑥) = 𝑦)
2623, 24, 253bitr4g 306 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ 𝑦 = (𝐴 𝑥)))
271, 7, 16, 26f1o2d 7152 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  cmpt 4954   × cxp 5344  wf 6123  1-1-ontowf1o 6126  cfv 6127  (class class class)co 6910  Basecbs 16229  Grpcgrp 17783  invgcminusg 17784   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-rep 4996  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-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  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-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-ga 18080
This theorem is referenced by:  galactghm  18180
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