Theorem gapm2 14732

Description: The action of a particular group element is a permutation of the base set. gapm 9462 expressed with the currying operator.

Hypotheses

Ref	Expression
gapm2.1	|- X = ran G
gapm2.2	|- Y = ran M

Assertion

Ref	Expression
gapm2	|- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> ((cur1` M)` A):Y-1-1-onto->Y)

Proof of Theorem gapm2

Step	Hyp	Ref	Expression
1		gagrp 9456	. . . . 5 |- ((M e. B /\ <.G, M>. e. GrpAct) -> G e. Grp)
2		gapm2.1	. . . . . 6 |- X = ran G
3	2	unsgrp 14728	. . . . 5 |- (G e. Grp -> X e. _V)
4	1, 3	syl 12	. . . 4 |- ((M e. B /\ <.G, M>. e. GrpAct) -> X e. _V)
5	4	3adant3 896	. . 3 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> X e. _V)
6		rnexg 4207	. . . . 5 |- (M e. B -> ran M e. _V)
7		gapm2.2	. . . . 5 |- Y = ran M
8	6, 7	syl5eqel 1975	. . . 4 |- (M e. B -> Y e. _V)
9	8	3ad2ant1 897	. . 3 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> Y e. _V)
10		simp3r 905	. . 3 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> Y =/= (/))
11	2, 7	gaf 9457	. . . . 5 |- ((M e. B /\ <.G, M>. e. GrpAct) -> M:(X X. Y)-->Y)
12		ffn 4562	. . . . 5 |- (M:(X X. Y)-->Y -> M Fn (X X. Y))
13	11, 12	syl 12	. . . 4 |- ((M e. B /\ <.G, M>. e. GrpAct) -> M Fn (X X. Y))
14	13	3adant3 896	. . 3 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> M Fn (X X. Y))
15		simp3l 904	. . 3 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> A e. X)
16		eqid 1884	. . . 4 |- (M o. `'(2nd |` ({A} X. _V))) = (M o. `'(2nd |` ({A} X. _V)))
17	16	valcurfn1 14552	. . 3 |- (((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ A e. X) -> ((cur1` M)` A) = (M o. `'(2nd |` ({A} X. _V))))
18	5, 9, 10, 14, 15, 17	syl221anc 1111	. 2 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> ((cur1` M)` A) = (M o. `'(2nd |` ({A} X. _V))))
19	2, 7, 16	gapm 9462	. . 3 |- ((M e. B /\ <.G, M>. e. GrpAct /\ A e. X) -> (M o. `'(2nd |` ({A} X. _V))):Y-1-1-onto->Y)
20	19	3adant3r 1095	. 2 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> (M o. `'(2nd |` ({A} X. _V))):Y-1-1-onto->Y)
21		f1oeq1 4630	. . . 4 |- ((M o. `'(2nd |` ({A} X. _V))) = ((cur1` M)` A) -> ((M o. `'(2nd |` ({A} X. _V))):Y-1-1-onto->Y <-> ((cur1` M)` A):Y-1-1-onto->Y))
22	21	biimpd 170	. . 3 |- ((M o. `'(2nd |` ({A} X. _V))) = ((cur1` M)` A) -> ((M o. `'(2nd |` ({A} X. _V))):Y-1-1-onto->Y -> ((cur1` M)` A):Y-1-1-onto->Y))
23	22	eqcoms 1887	. 2 |- (((cur1` M)` A) = (M o. `'(2nd |` ({A} X. _V))) -> ((M o. `'(2nd |` ({A} X. _V))):Y-1-1-onto->Y -> ((cur1` M)` A):Y-1-1-onto->Y))
24	18, 20, 23	sylc 83	1 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> ((cur1` M)` A):Y-1-1-onto->Y)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046   X. cxp 3984  `'ccnv 3985  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  2ndc2nd 5019  Grpcgr 9311  GrpActcga 9447  cur1ccur1 14542

This theorem is referenced by:  rngapm 14733  curgrpact 14735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-ga 9448  df-cur1 14544

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