Theorem wemappo 8454

Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
wemapso.t	|- T = { <. x , y >. | E. z e. A ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) }

Assertion

Ref	Expression
wemappo	|- ( ( A e. V /\ R Or A /\ S Po B ) -> T Po ( B ^m A ) )

Distinct variable groups:   x, B   x, w, y, z, A   w, R, x, y, z   w, S, x, y, z

Allowed substitution hints:   B( y, z, w)   T( x, y, z, w)   V( x, y, z, w)

Proof of Theorem wemappo

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		simpll3 1102	. . . . . . 7 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) /\ b e. A ) -> S Po B )
3		elmapi 7879	. . . . . . . . 9 |- ( a e. ( B ^m A ) -> a : A --> B )
4	3	adantl 482	. . . . . . . 8 |- ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) -> a : A --> B )
5	4	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) /\ b e. A ) -> ( a ` b ) e. B )
6		poirr 5046	. . . . . . 7 |- ( ( S Po B /\ ( a ` b ) e. B ) -> -. ( a ` b ) S ( a ` b ) )
7	2, 5, 6	syl2anc 693	. . . . . 6 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) /\ b e. A ) -> -. ( a ` b ) S ( a ` b ) )
8	7	intnanrd 963	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) /\ b e. A ) -> -. ( ( a ` b ) S ( a ` b ) /\ A. c e. A ( c R b -> ( a ` c ) = ( a ` c ) ) ) )
9	8	nrexdv 3001	. . . 4 |- ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) -> -. E. b e. A ( ( a ` b ) S ( a ` b ) /\ A. c e. A ( c R b -> ( a ` c ) = ( a ` c ) ) ) )
10		vex 3203	. . . . 5 |- a e. _V
11		wemapso.t	. . . . . 6 |- T = { <. x , y >. | E. z e. A ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) }
12	11	wemaplem1 8451	. . . . 5 |- ( ( a e. _V /\ a e. _V ) -> ( a T a <-> E. b e. A ( ( a ` b ) S ( a ` b ) /\ A. c e. A ( c R b -> ( a ` c ) = ( a ` c ) ) ) ) )
13	10, 10, 12	mp2an 708	. . . 4 |- ( a T a <-> E. b e. A ( ( a ` b ) S ( a ` b ) /\ A. c e. A ( c R b -> ( a ` c ) = ( a ` c ) ) ) )
14	9, 13	sylnibr 319	. . 3 |- ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ a e. ( B ^m A ) ) -> -. a T a )
15		simpll1 1100	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> A e. _V )
16		simplr1 1103	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> a e. ( B ^m A ) )
17		simplr2 1104	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> b e. ( B ^m A ) )
18		simplr3 1105	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> c e. ( B ^m A ) )
19		simpll2 1101	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> R Or A )
20		simpll3 1102	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> S Po B )
21		simprl 794	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> a T b )
22		simprr 796	. . . . 5 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> b T c )
23	11, 15, 16, 17, 18, 19, 20, 21, 22	wemaplem3 8453	. . . 4 |- ( ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) /\ ( a T b /\ b T c ) ) -> a T c )
24	23	ex 450	. . 3 |- ( ( ( A e. _V /\ R Or A /\ S Po B ) /\ ( a e. ( B ^m A ) /\ b e. ( B ^m A ) /\ c e. ( B ^m A ) ) ) -> ( ( a T b /\ b T c ) -> a T c ) )
25	14, 24	ispod 5043	. 2 |- ( ( A e. _V /\ R Or A /\ S Po B ) -> T Po ( B ^m A ) )
26	1, 25	syl3an1 1359	1 |- ( ( A e. V /\ R Or A /\ S Po B ) -> T Po ( B ^m A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   {copab 4712   Po wpo 5033   Or wor 5034   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-3or 1038  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-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-po 5035  df-so 5036  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  wemapsolem  8455