Theorem erdszelem1 31173

Description: Lemma for erdsze 31184. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypothesis

Ref	Expression
erdszelem1.1	|- S = { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) }

Assertion

Ref	Expression
erdszelem1	|- ( X e. S <-> ( X C_ ( 1 ... A ) /\ ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) )

Distinct variable groups:   y, A   y, F   y, O   y, X

Allowed substitution hint:   S( y)

Proof of Theorem erdszelem1

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( 1 ... A ) e. _V
2	1	elpw2 4828	. . 3 |- ( X e. ~P ( 1 ... A ) <-> X C_ ( 1 ... A ) )
3	2	anbi1i 731	. 2 |- ( ( X e. ~P ( 1 ... A ) /\ ( ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) ) <-> ( X C_ ( 1 ... A ) /\ ( ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) ) )
4		reseq2 5391	. . . . . 6 |- ( y = X -> ( F |` y ) = ( F |` X ) )
5		isoeq1 6567	. . . . . 6 |- ( ( F |` y ) = ( F |` X ) -> ( ( F |` y ) Isom < , O ( y , ( F " y ) ) <-> ( F |` X ) Isom < , O ( y , ( F " y ) ) ) )
6	4, 5	syl 17	. . . . 5 |- ( y = X -> ( ( F |` y ) Isom < , O ( y , ( F " y ) ) <-> ( F |` X ) Isom < , O ( y , ( F " y ) ) ) )
7		isoeq4 6570	. . . . 5 |- ( y = X -> ( ( F |` X ) Isom < , O ( y , ( F " y ) ) <-> ( F |` X ) Isom < , O ( X , ( F " y ) ) ) )
8		imaeq2 5462	. . . . . 6 |- ( y = X -> ( F " y ) = ( F " X ) )
9		isoeq5 6571	. . . . . 6 |- ( ( F " y ) = ( F " X ) -> ( ( F |` X ) Isom < , O ( X , ( F " y ) ) <-> ( F |` X ) Isom < , O ( X , ( F " X ) ) ) )
10	8, 9	syl 17	. . . . 5 |- ( y = X -> ( ( F |` X ) Isom < , O ( X , ( F " y ) ) <-> ( F |` X ) Isom < , O ( X , ( F " X ) ) ) )
11	6, 7, 10	3bitrd 294	. . . 4 |- ( y = X -> ( ( F |` y ) Isom < , O ( y , ( F " y ) ) <-> ( F |` X ) Isom < , O ( X , ( F " X ) ) ) )
12		eleq2 2690	. . . 4 |- ( y = X -> ( A e. y <-> A e. X ) )
13	11, 12	anbi12d 747	. . 3 |- ( y = X -> ( ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) <-> ( ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) ) )
14		erdszelem1.1	. . 3 |- S = { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) }
15	13, 14	elrab2 3366	. 2 |- ( X e. S <-> ( X e. ~P ( 1 ... A ) /\ ( ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) ) )
16		3anass 1042	. 2 |- ( ( X C_ ( 1 ... A ) /\ ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) <-> ( X C_ ( 1 ... A ) /\ ( ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) ) )
17	3, 15, 16	3bitr4i 292	1 |- ( X e. S <-> ( X C_ ( 1 ... A ) /\ ( F |` X ) Isom < , O ( X , ( F " X ) ) /\ A e. X ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   |` cres 5116   "cima 5117   Isom wiso 5889  (class class class)co 6650   1c1 9937   < clt 10074   ...cfz 12326

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-ov 6653

This theorem is referenced by:  erdszelem2  31174  erdszelem4  31176  erdszelem7  31179  erdszelem8  31180