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Theorem gapm 19336
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 19325 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
43ad2antrr 726 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → :(𝑋 × 𝑌)⟶𝑌)
5 simplr 769 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝐴𝑋)
6 simpr 484 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝑥𝑌)
74, 5, 6fovcdmd 7604 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → (𝐴 𝑥) ∈ 𝑌)
83ad2antrr 726 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → :(𝑋 × 𝑌)⟶𝑌)
9 gagrp 19322 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 726 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐺 ∈ Grp)
11 simplr 769 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐴𝑋)
12 eqid 2734 . . . . 5 (invg𝐺) = (invg𝐺)
132, 12grpinvcl 19017 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
1410, 11, 13syl2anc 584 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr 484 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
168, 14, 15fovcdmd 7604 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → (((invg𝐺)‘𝐴) 𝑦) ∈ 𝑌)
17 simpll 767 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ∈ (𝐺 GrpAct 𝑌))
18 simplr 769 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝐴𝑋)
19 simprl 771 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
20 simprr 773 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
212, 12gacan 19335 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2217, 18, 19, 20, 21syl13anc 1371 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2322bicomd 223 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((((invg𝐺)‘𝐴) 𝑦) = 𝑥 ↔ (𝐴 𝑥) = 𝑦))
24 eqcom 2741 . . 3 (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥)
25 eqcom 2741 . . 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 7686 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  cmpt 5230   × cxp 5686  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Basecbs 17244  Grpcgrp 18963  invgcminusg 18964   GrpAct cga 19319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-ga 19320
This theorem is referenced by:  galactghm  19436
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