Theorem f1omvdmvd 17863

Description: A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
f1omvdmvd	|- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( F ` X ) e. ( dom ( F \ _I ) \ { X } ) )

Proof of Theorem f1omvdmvd

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> X e. dom ( F \ _I ) )
2		f1ofn 6138	. . . . . . 7 |- ( F : A -1-1-onto-> A -> F Fn A )
3	2	adantr 481	. . . . . 6 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> F Fn A )
4		difss 3737	. . . . . . . . 9 |- ( F \ _I ) C_ F
5		dmss 5323	. . . . . . . . 9 |- ( ( F \ _I ) C_ F -> dom ( F \ _I ) C_ dom F )
6	4, 5	ax-mp 5	. . . . . . . 8 |- dom ( F \ _I ) C_ dom F
7		f1odm 6141	. . . . . . . 8 |- ( F : A -1-1-onto-> A -> dom F = A )
8	6, 7	syl5sseq 3653	. . . . . . 7 |- ( F : A -1-1-onto-> A -> dom ( F \ _I ) C_ A )
9	8	sselda 3603	. . . . . 6 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> X e. A )
10		fnelnfp 6443	. . . . . 6 |- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F \ _I ) <-> ( F ` X ) =/= X ) )
11	3, 9, 10	syl2anc 693	. . . . 5 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( X e. dom ( F \ _I ) <-> ( F ` X ) =/= X ) )
12	1, 11	mpbid 222	. . . 4 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( F ` X ) =/= X )
13		f1of1 6136	. . . . . . 7 |- ( F : A -1-1-onto-> A -> F : A -1-1-> A )
14	13	adantr 481	. . . . . 6 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> F : A -1-1-> A )
15		f1of 6137	. . . . . . . 8 |- ( F : A -1-1-onto-> A -> F : A --> A )
16	15	adantr 481	. . . . . . 7 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> F : A --> A )
17	16, 9	ffvelrnd 6360	. . . . . 6 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( F ` X ) e. A )
18		f1fveq 6519	. . . . . 6 |- ( ( F : A -1-1-> A /\ ( ( F ` X ) e. A /\ X e. A ) ) -> ( ( F ` ( F ` X ) ) = ( F ` X ) <-> ( F ` X ) = X ) )
19	14, 17, 9, 18	syl12anc 1324	. . . . 5 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( ( F ` ( F ` X ) ) = ( F ` X ) <-> ( F ` X ) = X ) )
20	19	necon3bid 2838	. . . 4 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( ( F ` ( F ` X ) ) =/= ( F ` X ) <-> ( F ` X ) =/= X ) )
21	12, 20	mpbird 247	. . 3 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( F ` ( F ` X ) ) =/= ( F ` X ) )
22		fnelnfp 6443	. . . 4 |- ( ( F Fn A /\ ( F ` X ) e. A ) -> ( ( F ` X ) e. dom ( F \ _I ) <-> ( F ` ( F ` X ) ) =/= ( F ` X ) ) )
23	3, 17, 22	syl2anc 693	. . 3 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( ( F ` X ) e. dom ( F \ _I ) <-> ( F ` ( F ` X ) ) =/= ( F ` X ) ) )
24	21, 23	mpbird 247	. 2 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( F ` X ) e. dom ( F \ _I ) )
25		eldifsn 4317	. 2 |- ( ( F ` X ) e. ( dom ( F \ _I ) \ { X } ) <-> ( ( F ` X ) e. dom ( F \ _I ) /\ ( F ` X ) =/= X ) )
26	24, 12, 25	sylanbrc 698	1 |- ( ( F : A -1-1-onto-> A /\ X e. dom ( F \ _I ) ) -> ( F ` X ) e. ( dom ( F \ _I ) \ { X } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   _I cid 5023   dom cdm 5114   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-f1o 5895  df-fv 5896

This theorem is referenced by:  f1otrspeq  17867  symggen  17890