Theorem ltbval 19471

Description: Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
ltbval.c	|- C = ( T <bag I )
ltbval.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
ltbval.i	|- ( ph -> I e. V )
ltbval.t	|- ( ph -> T e. W )

Assertion

Ref	Expression
ltbval	|- ( ph -> C = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } )

Distinct variable groups:   x, y, D   w, h, x, y, z, I   ph, h, x, y   w, T, x, y, z

Allowed substitution hints:   ph( z, w)   C( x, y, z, w, h)   D( z, w, h)   T( h)   V( x, y, z, w, h)   W( x, y, z, w, h)

Proof of Theorem ltbval

Dummy variables i r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltbval.c	. 2 |- C = ( T <bag I )
2		ltbval.t	. . 3 |- ( ph -> T e. W )
3		ltbval.i	. . 3 |- ( ph -> I e. V )
4		elex 3212	. . . 4 |- ( T e. W -> T e. _V )
5		elex 3212	. . . 4 |- ( I e. V -> I e. _V )
6		simpr 477	. . . . . . . . . . 11 |- ( ( r = T /\ i = I ) -> i = I )
7	6	oveq2d 6666	. . . . . . . . . 10 |- ( ( r = T /\ i = I ) -> ( NN0 ^m i ) = ( NN0 ^m I ) )
8		rabeq 3192	. . . . . . . . . 10 |- ( ( NN0 ^m i ) = ( NN0 ^m I ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
9	7, 8	syl 17	. . . . . . . . 9 |- ( ( r = T /\ i = I ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
10		ltbval.d	. . . . . . . . 9 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
11	9, 10	syl6eqr 2674	. . . . . . . 8 |- ( ( r = T /\ i = I ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = D )
12	11	sseq2d 3633	. . . . . . 7 |- ( ( r = T /\ i = I ) -> ( { x , y } C_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } <-> { x , y } C_ D ) )
13		simpl 473	. . . . . . . . . . . 12 |- ( ( r = T /\ i = I ) -> r = T )
14	13	breqd 4664	. . . . . . . . . . 11 |- ( ( r = T /\ i = I ) -> ( z r w <-> z T w ) )
15	14	imbi1d 331	. . . . . . . . . 10 |- ( ( r = T /\ i = I ) -> ( ( z r w -> ( x ` w ) = ( y ` w ) ) <-> ( z T w -> ( x ` w ) = ( y ` w ) ) ) )
16	6, 15	raleqbidv 3152	. . . . . . . . 9 |- ( ( r = T /\ i = I ) -> ( A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) <-> A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) )
17	16	anbi2d 740	. . . . . . . 8 |- ( ( r = T /\ i = I ) -> ( ( ( x ` z ) < ( y ` z ) /\ A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) ) <-> ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) )
18	6, 17	rexeqbidv 3153	. . . . . . 7 |- ( ( r = T /\ i = I ) -> ( E. z e. i ( ( x ` z ) < ( y ` z ) /\ A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) ) <-> E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) )
19	12, 18	anbi12d 747	. . . . . 6 |- ( ( r = T /\ i = I ) -> ( ( { x , y } C_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } /\ E. z e. i ( ( x ` z ) < ( y ` z ) /\ A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) ) ) <-> ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) ) )
20	19	opabbidv 4716	. . . . 5 |- ( ( r = T /\ i = I ) -> { <. x , y >. | ( { x , y } C_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } /\ E. z e. i ( ( x ` z ) < ( y ` z ) /\ A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) ) ) } = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } )
21		df-ltbag 19359	. . . . 5 |- <bag = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } /\ E. z e. i ( ( x ` z ) < ( y ` z ) /\ A. w e. i ( z r w -> ( x ` w ) = ( y ` w ) ) ) ) } )
22		vex 3203	. . . . . . . . 9 |- x e. _V
23		vex 3203	. . . . . . . . 9 |- y e. _V
24	22, 23	prss 4351	. . . . . . . 8 |- ( ( x e. D /\ y e. D ) <-> { x , y } C_ D )
25	24	anbi1i 731	. . . . . . 7 |- ( ( ( x e. D /\ y e. D ) /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) <-> ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) )
26	25	opabbii 4717	. . . . . 6 |- { <. x , y >. | ( ( x e. D /\ y e. D ) /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) }
27		ovex 6678	. . . . . . . . 9 |- ( NN0 ^m I ) e. _V
28	10, 27	rabex2 4815	. . . . . . . 8 |- D e. _V
29	28, 28	xpex 6962	. . . . . . 7 |- ( D X. D ) e. _V
30		opabssxp 5193	. . . . . . 7 |- { <. x , y >. | ( ( x e. D /\ y e. D ) /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } C_ ( D X. D )
31	29, 30	ssexi 4803	. . . . . 6 |- { <. x , y >. | ( ( x e. D /\ y e. D ) /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } e. _V
32	26, 31	eqeltrri 2698	. . . . 5 |- { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } e. _V
33	20, 21, 32	ovmpt2a 6791	. . . 4 |- ( ( T e. _V /\ I e. _V ) -> ( T <bag I ) = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } )
34	4, 5, 33	syl2an 494	. . 3 |- ( ( T e. W /\ I e. V ) -> ( T <bag I ) = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } )
35	2, 3, 34	syl2anc 693	. 2 |- ( ph -> ( T <bag I ) = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } )
36	1, 35	syl5eq 2668	1 |- ( ph -> C = { <. x , y >. | ( { x , y } C_ D /\ E. z e. I ( ( x ` z ) < ( y ` z ) /\ A. w e. I ( z T w -> ( x ` w ) = ( y ` w ) ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   {copab 4712   X. cxp 5112   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   < clt 10074   NNcn 11020   NN0cn0 11292   <_bag cltb 19354

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-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-oprab 6654  df-mpt2 6655  df-ltbag 19359

This theorem is referenced by:  ltbwe  19472