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Theorem gapm 19170
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 19159 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
43ad2antrr 725 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → :(𝑋 × 𝑌)⟶𝑌)
5 simplr 768 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝐴𝑋)
6 simpr 486 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝑥𝑌)
74, 5, 6fovcdmd 7579 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → (𝐴 𝑥) ∈ 𝑌)
83ad2antrr 725 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → :(𝑋 × 𝑌)⟶𝑌)
9 gagrp 19156 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 725 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐺 ∈ Grp)
11 simplr 768 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐴𝑋)
12 eqid 2733 . . . . 5 (invg𝐺) = (invg𝐺)
132, 12grpinvcl 18872 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
1410, 11, 13syl2anc 585 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr 486 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
168, 14, 15fovcdmd 7579 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → (((invg𝐺)‘𝐴) 𝑦) ∈ 𝑌)
17 simpll 766 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ∈ (𝐺 GrpAct 𝑌))
18 simplr 768 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝐴𝑋)
19 simprl 770 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
20 simprr 772 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
212, 12gacan 19169 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2217, 18, 19, 20, 21syl13anc 1373 . . . 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 2740 . . 3 (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥)
25 eqcom 2740 . . 3 (𝑦 = (𝐴 𝑥) ↔ (𝐴 𝑥) = 𝑦)
2623, 24, 253bitr4g 314 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ 𝑦 = (𝐴 𝑥)))
271, 7, 16, 26f1o2d 7660 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cmpt 5232   × cxp 5675  wf 6540  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  Basecbs 17144  Grpcgrp 18819  invgcminusg 18820   GrpAct cga 19153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-ga 19154
This theorem is referenced by:  galactghm  19272
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