Theorem suplub2 8367

Description: Bidirectional form of suplub 8366. (Contributed by Mario Carneiro, 6-Sep-2014.)

Hypotheses

Ref	Expression
supmo.1	|- ( ph -> R Or A )
supcl.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 ) ) )
suplub2.3	|- ( ph -> B C_ A )

Assertion

Ref	Expression
suplub2	|- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) <-> E. z e. B C R z ) )

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

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

Proof of Theorem suplub2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmo.1	. . . 4 |- ( ph -> R Or A )
2		supcl.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	suplub 8366	. . 3 |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) -> E. z e. B C R z ) )
4	3	expdimp 453	. 2 |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) -> E. z e. B C R z ) )
5		breq2 4657	. . . 4 |- ( z = w -> ( C R z <-> C R w ) )
6	5	cbvrexv 3172	. . 3 |- ( E. z e. B C R z <-> E. w e. B C R w )
7		breq2 4657	. . . . . . 7 |- ( sup ( B , A , R ) = w -> ( C R sup ( B , A , R ) <-> C R w ) )
8	7	biimprd 238	. . . . . 6 |- ( sup ( B , A , R ) = w -> ( C R w -> C R sup ( B , A , R ) ) )
9	8	a1i 11	. . . . 5 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( sup ( B , A , R ) = w -> ( C R w -> C R sup ( B , A , R ) ) ) )
10	1	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> R Or A )
11		simplr 792	. . . . . . 7 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> C e. A )
12		suplub2.3	. . . . . . . . 9 |- ( ph -> B C_ A )
13	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ C e. A ) -> B C_ A )
14	13	sselda 3603	. . . . . . 7 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> w e. A )
15	1, 2	supcl 8364	. . . . . . . 8 |- ( ph -> sup ( B , A , R ) e. A )
16	15	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> sup ( B , A , R ) e. A )
17		sotr 5057	. . . . . . 7 |- ( ( R Or A /\ ( C e. A /\ w e. A /\ sup ( B , A , R ) e. A ) ) -> ( ( C R w /\ w R sup ( B , A , R ) ) -> C R sup ( B , A , R ) ) )
18	10, 11, 14, 16, 17	syl13anc 1328	. . . . . 6 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( ( C R w /\ w R sup ( B , A , R ) ) -> C R sup ( B , A , R ) ) )
19	18	expcomd 454	. . . . 5 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( w R sup ( B , A , R ) -> ( C R w -> C R sup ( B , A , R ) ) ) )
20	1, 2	supub 8365	. . . . . . . 8 |- ( ph -> ( w e. B -> -. sup ( B , A , R ) R w ) )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ C e. A ) -> ( w e. B -> -. sup ( B , A , R ) R w ) )
22	21	imp 445	. . . . . 6 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> -. sup ( B , A , R ) R w )
23		sotric 5061	. . . . . . . 8 |- ( ( R Or A /\ ( sup ( B , A , R ) e. A /\ w e. A ) ) -> ( sup ( B , A , R ) R w <-> -. ( sup ( B , A , R ) = w \/ w R sup ( B , A , R ) ) ) )
24	10, 16, 14, 23	syl12anc 1324	. . . . . . 7 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( sup ( B , A , R ) R w <-> -. ( sup ( B , A , R ) = w \/ w R sup ( B , A , R ) ) ) )
25	24	con2bid 344	. . . . . 6 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( ( sup ( B , A , R ) = w \/ w R sup ( B , A , R ) ) <-> -. sup ( B , A , R ) R w ) )
26	22, 25	mpbird 247	. . . . 5 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( sup ( B , A , R ) = w \/ w R sup ( B , A , R ) ) )
27	9, 19, 26	mpjaod 396	. . . 4 |- ( ( ( ph /\ C e. A ) /\ w e. B ) -> ( C R w -> C R sup ( B , A , R ) ) )
28	27	rexlimdva 3031	. . 3 |- ( ( ph /\ C e. A ) -> ( E. w e. B C R w -> C R sup ( B , A , R ) ) )
29	6, 28	syl5bi 232	. 2 |- ( ( ph /\ C e. A ) -> ( E. z e. B C R z -> C R sup ( B , A , R ) ) )
30	4, 29	impbid 202	1 |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) <-> E. z e. B C R z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   Or wor 5034   supcsup 8346

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  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-op 4184  df-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-iota 5851  df-riota 6611  df-sup 8348

This theorem is referenced by:  infglbb  8397  suprlub  10987  supxrlub  12155