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Theorem gapm 19286
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 19275 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
43ad2antrr 724 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → :(𝑋 × 𝑌)⟶𝑌)
5 simplr 767 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝐴𝑋)
6 simpr 483 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝑥𝑌)
74, 5, 6fovcdmd 7593 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → (𝐴 𝑥) ∈ 𝑌)
83ad2antrr 724 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → :(𝑋 × 𝑌)⟶𝑌)
9 gagrp 19272 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 724 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐺 ∈ Grp)
11 simplr 767 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐴𝑋)
12 eqid 2725 . . . . 5 (invg𝐺) = (invg𝐺)
132, 12grpinvcl 18968 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
1410, 11, 13syl2anc 582 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr 483 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
168, 14, 15fovcdmd 7593 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → (((invg𝐺)‘𝐴) 𝑦) ∈ 𝑌)
17 simpll 765 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ∈ (𝐺 GrpAct 𝑌))
18 simplr 767 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝐴𝑋)
19 simprl 769 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
20 simprr 771 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
212, 12gacan 19285 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2217, 18, 19, 20, 21syl13anc 1369 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2322bicomd 222 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((((invg𝐺)‘𝐴) 𝑦) = 𝑥 ↔ (𝐴 𝑥) = 𝑦))
24 eqcom 2732 . . 3 (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥)
25 eqcom 2732 . . 3 (𝑦 = (𝐴 𝑥) ↔ (𝐴 𝑥) = 𝑦)
2623, 24, 253bitr4g 313 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ 𝑦 = (𝐴 𝑥)))
271, 7, 16, 26f1o2d 7675 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cmpt 5232   × cxp 5676  wf 6545  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  Basecbs 17199  Grpcgrp 18914  invgcminusg 18915   GrpAct cga 19269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-0g 17442  df-mgm 18619  df-sgrp 18698  df-mnd 18714  df-grp 18917  df-minusg 18918  df-ga 19270
This theorem is referenced by:  galactghm  19388
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