Theorem dihopelvalbN 36527

Description: Ordered pair member of the partial isomorphism H for argument under W. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihval3.b	|- B = ( Base ` K )
dihval3.l	|- .<_ = ( le ` K )
dihval3.h	|- H = ( LHyp ` K )
dihval3.t	|- T = ( ( LTrn ` K ) ` W )
dihval3.r	|- R = ( ( trL ` K ) ` W )
dihval3.o	|- O = ( g e. T |-> ( _I |` B ) )
dihval3.i	|- I = ( ( DIsoH ` K ) ` W )

Assertion

Ref	Expression
dihopelvalbN	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = O ) ) )

Distinct variable groups:   g, K   T, g   g, W

Allowed substitution hints:   B( g)   R( g)   S( g)   F( g)   H( g)   I( g)   .<_ ( g)   O( g)   V( g)   X( g)

Proof of Theorem dihopelvalbN

Step	Hyp	Ref	Expression
1		dihval3.b	. . . 4 |- B = ( Base ` K )
2		dihval3.l	. . . 4 |- .<_ = ( le ` K )
3		dihval3.h	. . . 4 |- H = ( LHyp ` K )
4		dihval3.i	. . . 4 |- I = ( ( DIsoH ` K ) ` W )
5		eqid 2622	. . . 4 |- ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
6	1, 2, 3, 4, 5	dihvalb 36526	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) )
7	6	eleq2d 2687	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> <. F , S >. e. ( ( ( DIsoB ` K ) ` W ) ` X ) ) )
8		dihval3.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
9		dihval3.r	. . 3 |- R = ( ( trL ` K ) ` W )
10		dihval3.o	. . 3 |- O = ( g e. T |-> ( _I |` B ) )
11	1, 2, 3, 8, 9, 10, 5	dibopelval3 36437	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( ( ( DIsoB ` K ) ` W ) ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = O ) ) )
12	7, 11	bitrd 268	1 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = O ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   DIsoBcdib 36427   DIsoHcdih 36517

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-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-riota 6611  df-ov 6653  df-disoa 36318  df-dib 36428  df-dih 36518

This theorem is referenced by:  dihmeetlem1N  36579  dihglblem5apreN  36580  dihmeetlem4preN  36595