Theorem suplub2ti 6509

Description: Bidirectional form of suplubti 6508. (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 6508	. . 3 |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) -> E. z e. B C R z ) )
4	3	expdimp 255	. 2 |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) -> E. z e. B C R z ) )
5		breq2 3810	. . . 4 |- ( z = w -> ( C R z <-> C R w ) )
6	5	cbvrexv 2583	. . 3 |- ( E. z e. B C R z <-> E. w e. B C R w )
7		simplll 500	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> ph )
8		simplr 497	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> w e. B )
9	1, 2	supubti 6507	. . . . . . 7 |- ( ph -> ( w e. B -> -. sup ( B , A , R ) R w ) )
10	7, 8, 9	sylc 61	. . . . . 6 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> -. sup ( B , A , R ) R w )
11		simpr 108	. . . . . . 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 476	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> R Or A )
14		simpllr 501	. . . . . . . 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 476	. . . . . . . . 9 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> B C_ A )
17	16, 8	sseldd 3010	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> w e. A )
18	1, 2	supclti 6506	. . . . . . . . 9 |- ( ph -> sup ( B , A , R ) e. A )
19	18	ad3antrrr 476	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> sup ( B , A , R ) e. A )
20		sowlin 4104	. . . . . . . 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 1172	. . . . . . 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 1281	. . . . 5 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> C R sup ( B , A , R ) )
24	23	ex 113	. . . 4 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( C R w -> C R sup ( B , A , R ) ) )
25	24	rexlimdva 2482	. . 3 |- ( ( ph /\ C e. A ) -> ( E. w e. B C R w -> C R sup ( B , A , R ) ) )
26	6, 25	syl5bi 150	. 2 |- ( ( ph /\ C e. A ) -> ( E. z e. B C R z -> C R sup ( B , A , R ) ) )
27	4, 26	impbid 127	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 102   <-> wb 103   \/ wo 662   e. wcel 1434   A.wral 2353   E.wrex 2354   C_ wss 2983   class class class wbr 3806   Or wor 4079   supcsup 6490

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-iso 4081  df-iota 4918  df-riota 5520  df-sup 6492

This theorem is referenced by:  suprlubex  8133