Theorem inftmrel 29734

Description: The infinitesimal relation for a structure W. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypothesis

Ref	Expression
inftm.b	|- B = ( Base ` W )

Assertion

Ref	Expression
inftmrel	|- ( W e. V -> (<<< ` W ) C_ ( B X. B ) )

Proof of Theorem inftmrel

Dummy variables x w y n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( W e. V -> W e. _V )
2		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
3		inftm.b	. . . . . . . . 9 |- B = ( Base ` W )
4	2, 3	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = B )
5	4	eleq2d 2687	. . . . . . 7 |- ( w = W -> ( x e. ( Base ` w ) <-> x e. B ) )
6	4	eleq2d 2687	. . . . . . 7 |- ( w = W -> ( y e. ( Base ` w ) <-> y e. B ) )
7	5, 6	anbi12d 747	. . . . . 6 |- ( w = W -> ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) <-> ( x e. B /\ y e. B ) ) )
8		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
9		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( lt ` w ) = ( lt ` W ) )
10		eqidd 2623	. . . . . . . 8 |- ( w = W -> x = x )
11	8, 9, 10	breq123d 4667	. . . . . . 7 |- ( w = W -> ( ( 0g ` w ) ( lt ` w ) x <-> ( 0g ` W ) ( lt ` W ) x ) )
12		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> (.g ` w ) = (.g ` W ) )
13	12	oveqd 6667	. . . . . . . . 9 |- ( w = W -> ( n (.g ` w ) x ) = ( n (.g ` W ) x ) )
14		eqidd 2623	. . . . . . . . 9 |- ( w = W -> y = y )
15	13, 9, 14	breq123d 4667	. . . . . . . 8 |- ( w = W -> ( ( n (.g ` w ) x ) ( lt ` w ) y <-> ( n (.g ` W ) x ) ( lt ` W ) y ) )
16	15	ralbidv 2986	. . . . . . 7 |- ( w = W -> ( A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y <-> A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) )
17	11, 16	anbi12d 747	. . . . . 6 |- ( w = W -> ( ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) <-> ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) )
18	7, 17	anbi12d 747	. . . . 5 |- ( w = W -> ( ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) <-> ( ( x e. B /\ y e. B ) /\ ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) ) )
19	18	opabbidv 4716	. . . 4 |- ( w = W -> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) } )
20		df-inftm 29732	. . . 4 |- <<< = ( w e. _V |-> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) } )
21		fvex 6201	. . . . . . 7 |- ( Base ` W ) e. _V
22	3, 21	eqeltri 2697	. . . . . 6 |- B e. _V
23	22, 22	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
24		opabssxp 5193	. . . . 5 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) } C_ ( B X. B )
25	23, 24	ssexi 4803	. . . 4 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) } e. _V
26	19, 20, 25	fvmpt 6282	. . 3 |- ( W e. _V -> (<<< ` W ) = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) } )
27	1, 26	syl 17	. 2 |- ( W e. V -> (<<< ` W ) = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( ( 0g ` W ) ( lt ` W ) x /\ A. n e. NN ( n (.g ` W ) x ) ( lt ` W ) y ) ) } )
28	27, 24	syl6eqss 3655	1 |- ( W e. V -> (<<< ` W ) C_ ( B X. B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   NNcn 11020   Basecbs 15857   0gc0g 16100   ltcplt 16941  ._gcmg 17540  <<<cinftm 29730

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-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-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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-inftm 29732

This theorem is referenced by:  isarchi  29736