Theorem suplub2ti 7076

Description: Bidirectional form of suplubti 7075. (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 7075	. . 3 |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) -> E. z e. B C R z ) )
4	3	expdimp 259	. 2 |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) -> E. z e. B C R z ) )
5		breq2 4038	. . . 4 |- ( z = w -> ( C R z <-> C R w ) )
6	5	cbvrexv 2730	. . 3 |- ( E. z e. B C R z <-> E. w e. B C R w )
7		simplll 533	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> ph )
8		simplr 528	. . . . . . 7 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> w e. B )
9	1, 2	supubti 7074	. . . . . . 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 110	. . . . . . 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 492	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> R Or A )
14		simpllr 534	. . . . . . . 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 492	. . . . . . . . 9 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> B C_ A )
17	16, 8	sseldd 3185	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> w e. A )
18	1, 2	supclti 7073	. . . . . . . . 9 |- ( ph -> sup ( B , A , R ) e. A )
19	18	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> sup ( B , A , R ) e. A )
20		sowlin 4356	. . . . . . . 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 1251	. . . . . . 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 1360	. . . . 5 |- ( ( ( ( ph /\ C e. A ) /\ w e. B ) /\ C R w ) -> C R sup ( B , A , R ) )
24	23	ex 115	. . . 4 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( C R w -> C R sup ( B , A , R ) ) )
25	24	rexlimdva 2614	. . 3 |- ( ( ph /\ C e. A ) -> ( E. w e. B C R w -> C R sup ( B , A , R ) ) )
26	6, 25	biimtrid 152	. 2 |- ( ( ph /\ C e. A ) -> ( E. z e. B C R z -> C R sup ( B , A , R ) ) )
27	4, 26	impbid 129	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 104   <-> wb 105   \/ wo 709   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   class class class wbr 4034   Or wor 4331   supcsup 7057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iso 4333  df-iota 5220  df-riota 5880  df-sup 7059

This theorem is referenced by:  suprlubex  8996