Theorem rngapm 14733

Description: The range of the action of a particular group element equals the range of the action.

Hypotheses

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

Assertion

Ref	Expression
rngapm	|- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> ran ((cur1` M)` A) = ran M)

Proof of Theorem rngapm

Step	Hyp	Ref	Expression
1		gapm2.1	. . 3 |- X = ran G
2		gapm2.2	. . 3 |- Y = ran M
3	1, 2	gapm2 14732	. 2 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> ((cur1` M)` A):Y-1-1-onto->Y)
4		f1ofo 4643	. . 3 |- (((cur1` M)` A):Y-1-1-onto->Y -> ((cur1` M)` A):Y-onto->Y)
5		forn 4620	. . 3 |- (((cur1` M)` A):Y-onto->Y -> ran ((cur1` M)` A) = Y)
6	2	eqeq2i 1894	. . . 4 |- (ran ((cur1` M)` A) = Y <-> ran ((cur1` M)` A) = ran M)
7	6	biimpi 168	. . 3 |- (ran ((cur1` M)` A) = Y -> ran ((cur1` M)` A) = ran M)
8	4, 5, 7	3syl 24	. 2 |- (((cur1` M)` A):Y-1-1-onto->Y -> ran ((cur1` M)` A) = ran M)
9	3, 8	syl 12	1 |- ((M e. B /\ <.G, M>. e. GrpAct /\ (A e. X /\ Y =/= (/))) -> ran ((cur1` M)` A) = ran M)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  (/)c0 2875  <.cop 3046  ran crn 3987  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  GrpActcga 9447  cur1ccur1 14542

This theorem is referenced by:  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

Copyright terms: Public domain