Theorem fnwe2 37623

Description: A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 7293 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
fnwe2.su	|- ( z = ( F ` x ) -> S = U )
fnwe2.t	|- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) }
fnwe2.s	|- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
fnwe2.f	|- ( ph -> ( F |` A ) : A --> B )
fnwe2.r	|- ( ph -> R We B )

Assertion

Ref	Expression
fnwe2	|- ( ph -> T We A )

Distinct variable groups:   y, U, z   x, S, y   x, R, y   ph, x, y, z   x, A, y, z   x, F, y, z

Allowed substitution hints:   B( x, y, z)   R( z)   S( z)   T( x, y, z)   U( x)

Proof of Theorem fnwe2

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

Step	Hyp	Ref	Expression
1		fnwe2.su	. . . . . 6 |- ( z = ( F ` x ) -> S = U )
2		fnwe2.t	. . . . . 6 |- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) }
3		fnwe2.s	. . . . . . 7 |- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
4	3	adantlr 751	. . . . . 6 |- ( ( ( ph /\ ( a C_ A /\ a =/= (/) ) ) /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
5		fnwe2.f	. . . . . . 7 |- ( ph -> ( F |` A ) : A --> B )
6	5	adantr 481	. . . . . 6 |- ( ( ph /\ ( a C_ A /\ a =/= (/) ) ) -> ( F |` A ) : A --> B )
7		fnwe2.r	. . . . . . 7 |- ( ph -> R We B )
8	7	adantr 481	. . . . . 6 |- ( ( ph /\ ( a C_ A /\ a =/= (/) ) ) -> R We B )
9		simprl 794	. . . . . 6 |- ( ( ph /\ ( a C_ A /\ a =/= (/) ) ) -> a C_ A )
10		simprr 796	. . . . . 6 |- ( ( ph /\ ( a C_ A /\ a =/= (/) ) ) -> a =/= (/) )
11	1, 2, 4, 6, 8, 9, 10	fnwe2lem2 37621	. . . . 5 |- ( ( ph /\ ( a C_ A /\ a =/= (/) ) ) -> E. c e. a A. d e. a -. d T c )
12	11	ex 450	. . . 4 |- ( ph -> ( ( a C_ A /\ a =/= (/) ) -> E. c e. a A. d e. a -. d T c ) )
13	12	alrimiv 1855	. . 3 |- ( ph -> A. a ( ( a C_ A /\ a =/= (/) ) -> E. c e. a A. d e. a -. d T c ) )
14		df-fr 5073	. . 3 |- ( T Fr A <-> A. a ( ( a C_ A /\ a =/= (/) ) -> E. c e. a A. d e. a -. d T c ) )
15	13, 14	sylibr 224	. 2 |- ( ph -> T Fr A )
16	3	adantlr 751	. . . 4 |- ( ( ( ph /\ ( a e. A /\ b e. A ) ) /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
17	5	adantr 481	. . . 4 |- ( ( ph /\ ( a e. A /\ b e. A ) ) -> ( F |` A ) : A --> B )
18	7	adantr 481	. . . 4 |- ( ( ph /\ ( a e. A /\ b e. A ) ) -> R We B )
19		simprl 794	. . . 4 |- ( ( ph /\ ( a e. A /\ b e. A ) ) -> a e. A )
20		simprr 796	. . . 4 |- ( ( ph /\ ( a e. A /\ b e. A ) ) -> b e. A )
21	1, 2, 16, 17, 18, 19, 20	fnwe2lem3 37622	. . 3 |- ( ( ph /\ ( a e. A /\ b e. A ) ) -> ( a T b \/ a = b \/ b T a ) )
22	21	ralrimivva 2971	. 2 |- ( ph -> A. a e. A A. b e. A ( a T b \/ a = b \/ b T a ) )
23		dfwe2 6981	. 2 |- ( T We A <-> ( T Fr A /\ A. a e. A A. b e. A ( a T b \/ a = b \/ b T a ) ) )
24	15, 22, 23	sylanbrc 698	1 |- ( ph -> T We A )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   {copab 4712   Fr wfr 5070   We wwe 5072   |` cres 5116   -->wf 5884   ` cfv 5888

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-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-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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

This theorem is referenced by:  aomclem4  37627