Theorem suplub2ti 6888

Description: Bidirectional form of suplubti 6887. (Contributed by Jim Kingdon, 17-Jan-2022.)

Hypotheses

Ref	Expression
supmoti.ti	|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
supclti.2	|- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
suplub2ti.or	|- ( ph -> R Or A )
suplub2ti.3	|- ( ph -> B C_ A )

Assertion

Ref	Expression
suplub2ti	|- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) <-> E. z e. B C R z ) )

Distinct variable groups:   u, A, v, x   y, A, x, z   x, B, y, z   u, R, v, x   y, R, z   ph, u, v, x   z, C

Allowed substitution hints:   ph( y, z)   B( v, u)   C( x, y, v, u)

Proof of Theorem suplub2ti

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmoti.ti	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
2		supclti.2	. . . 4 |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
3	1, 2	suplubti 6887	. . 3 |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) -> E. z e. B C R z ) )
4	3	expdimp 257	. 2 |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) -> E. z e. B C R z ) )
5		breq2 3933	. . . 4 |- ( z = w -> ( C R z <-> C R w ) )
6	5	cbvrexv 2655	. . 3 |- ( E. z e. B C R z <-> E. w e. B C R w )
7		simplll 522	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> ph )
8		simplr 519	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> w e. B )
9	1, 2	supubti 6886	. . . . . . 7 |- ( ph -> ( w e. B -> -. sup ( B , A , R ) R w ) )
10	7, 8, 9	sylc 62	. . . . . 6 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> -. sup ( B , A , R ) R w )
11		simpr 109	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> C R w )
12		suplub2ti.or	. . . . . . . . 9 |- ( ph -> R Or A )
13	12	ad3antrrr 483	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> R Or A )
14		simpllr 523	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> C e. A )
15		suplub2ti.3	. . . . . . . . . 10 |- ( ph -> B C_ A )
16	15	ad3antrrr 483	. . . . . . . . 9 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> B C_ A )
17	16, 8	sseldd 3098	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> w e. A )
18	1, 2	supclti 6885	. . . . . . . . 9 |- ( ph -> sup ( B , A , R ) e. A )
19	18	ad3antrrr 483	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> sup ( B , A , R ) e. A )
20		sowlin 4242	. . . . . . . 8 |- ( ( R Or A /\ ( C e. A /\ w e. A /\ sup ( B , A , R ) e. A ) ) -> ( C R w -> ( C R sup ( B , A , R ) \/ sup ( B , A , R ) R w ) ) )
21	13, 14, 17, 19, 20	syl13anc 1218	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> ( C R w -> ( C R sup ( B , A , R ) \/ sup ( B , A , R ) R w ) ) )
22	11, 21	mpd 13	. . . . . 6 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> ( C R sup ( B , A , R ) \/ sup ( B , A , R ) R w ) )
23	10, 22	ecased 1327	. . . . 5 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> C R sup ( B , A , R ) )
24	23	ex 114	. . . 4 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( C R w -> C R sup ( B , A , R ) ) )
25	24	rexlimdva 2549	. . 3 |- ( ( ph /\ C e. A ) -> ( E. w e. B C R w -> C R sup ( B , A , R ) ) )
26	6, 25	syl5bi 151	. 2 |- ( ( ph /\ C e. A ) -> ( E. z e. B C R z -> C R sup ( B , A , R ) ) )
27	4, 26	impbid 128	1 |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) <-> E. z e. B C R z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   e. wcel 1480   A.wral 2416   E.wrex 2417   C_ wss 3071   class class class wbr 3929   Or wor 4217   supcsup 6869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iso 4219  df-iota 5088  df-riota 5730  df-sup 6871

This theorem is referenced by:  suprlubex  8710