Theorem icccmplem1 22625

Description: Lemma for icccmp 22628. (Contributed by Mario Carneiro, 18-Jun-2014.)

Hypotheses

Ref	Expression
icccmp.1	|- J = ( topGen ` ran (,) )
icccmp.2	|- T = ( J ↾t ( A [,] B ) )
icccmp.3	|- D = ( ( abs o. - ) |` ( RR X. RR ) )
icccmp.4	|- S = { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z }
icccmp.5	|- ( ph -> A e. RR )
icccmp.6	|- ( ph -> B e. RR )
icccmp.7	|- ( ph -> A <_ B )
icccmp.8	|- ( ph -> U C_ J )
icccmp.9	|- ( ph -> ( A [,] B ) C_ U. U )

Assertion

Ref	Expression
icccmplem1	|- ( ph -> ( A e. S /\ A. y e. S y <_ B ) )

Distinct variable groups:   x, y, z, B   ph, y   x, A, y, z   x, D   x, T, z   z, J   y, S   x, U, y, z

Allowed substitution hints:   ph( x, z)   D( y, z)   S( x, z)   T( y)   J( x, y)

Proof of Theorem icccmplem1

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icccmp.5	. . . . 5 |- ( ph -> A e. RR )
2	1	rexrd 10089	. . . 4 |- ( ph -> A e. RR* )
3		icccmp.6	. . . . 5 |- ( ph -> B e. RR )
4	3	rexrd 10089	. . . 4 |- ( ph -> B e. RR* )
5		icccmp.7	. . . 4 |- ( ph -> A <_ B )
6		lbicc2 12288	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
7	2, 4, 5, 6	syl3anc 1326	. . 3 |- ( ph -> A e. ( A [,] B ) )
8		icccmp.9	. . . . . 6 |- ( ph -> ( A [,] B ) C_ U. U )
9	8, 7	sseldd 3604	. . . . 5 |- ( ph -> A e. U. U )
10		eluni2 4440	. . . . 5 |- ( A e. U. U <-> E. u e. U A e. u )
11	9, 10	sylib 208	. . . 4 |- ( ph -> E. u e. U A e. u )
12		snssi 4339	. . . . . . . 8 |- ( u e. U -> { u } C_ U )
13	12	ad2antrl 764	. . . . . . 7 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } C_ U )
14		snex 4908	. . . . . . . 8 |- { u } e. _V
15	14	elpw 4164	. . . . . . 7 |- ( { u } e. ~P U <-> { u } C_ U )
16	13, 15	sylibr 224	. . . . . 6 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } e. ~P U )
17		snfi 8038	. . . . . . 7 |- { u } e. Fin
18	17	a1i 11	. . . . . 6 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } e. Fin )
19	16, 18	elind 3798	. . . . 5 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { u } e. ( ~P U i^i Fin ) )
20	2	adantr 481	. . . . . . 7 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> A e. RR* )
21		iccid 12220	. . . . . . 7 |- ( A e. RR* -> ( A [,] A ) = { A } )
22	20, 21	syl 17	. . . . . 6 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> ( A [,] A ) = { A } )
23		snssi 4339	. . . . . . 7 |- ( A e. u -> { A } C_ u )
24	23	ad2antll 765	. . . . . 6 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> { A } C_ u )
25	22, 24	eqsstrd 3639	. . . . 5 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> ( A [,] A ) C_ u )
26		unieq 4444	. . . . . . . 8 |- ( z = { u } -> U. z = U. { u } )
27		vex 3203	. . . . . . . . 9 |- u e. _V
28	27	unisn 4451	. . . . . . . 8 |- U. { u } = u
29	26, 28	syl6eq 2672	. . . . . . 7 |- ( z = { u } -> U. z = u )
30	29	sseq2d 3633	. . . . . 6 |- ( z = { u } -> ( ( A [,] A ) C_ U. z <-> ( A [,] A ) C_ u ) )
31	30	rspcev 3309	. . . . 5 |- ( ( { u } e. ( ~P U i^i Fin ) /\ ( A [,] A ) C_ u ) -> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z )
32	19, 25, 31	syl2anc 693	. . . 4 |- ( ( ph /\ ( u e. U /\ A e. u ) ) -> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z )
33	11, 32	rexlimddv 3035	. . 3 |- ( ph -> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z )
34		oveq2 6658	. . . . . 6 |- ( x = A -> ( A [,] x ) = ( A [,] A ) )
35	34	sseq1d 3632	. . . . 5 |- ( x = A -> ( ( A [,] x ) C_ U. z <-> ( A [,] A ) C_ U. z ) )
36	35	rexbidv 3052	. . . 4 |- ( x = A -> ( E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z <-> E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) )
37		icccmp.4	. . . 4 |- S = { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z }
38	36, 37	elrab2 3366	. . 3 |- ( A e. S <-> ( A e. ( A [,] B ) /\ E. z e. ( ~P U i^i Fin ) ( A [,] A ) C_ U. z ) )
39	7, 33, 38	sylanbrc 698	. 2 |- ( ph -> A e. S )
40		ssrab2 3687	. . . . . 6 |- { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z } C_ ( A [,] B )
41	37, 40	eqsstri 3635	. . . . 5 |- S C_ ( A [,] B )
42	41	sseli 3599	. . . 4 |- ( y e. S -> y e. ( A [,] B ) )
43		elicc2 12238	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
44	1, 3, 43	syl2anc 693	. . . . . 6 |- ( ph -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
45	44	biimpa 501	. . . . 5 |- ( ( ph /\ y e. ( A [,] B ) ) -> ( y e. RR /\ A <_ y /\ y <_ B ) )
46	45	simp3d 1075	. . . 4 |- ( ( ph /\ y e. ( A [,] B ) ) -> y <_ B )
47	42, 46	sylan2 491	. . 3 |- ( ( ph /\ y e. S ) -> y <_ B )
48	47	ralrimiva 2966	. 2 |- ( ph -> A. y e. S y <_ B )
49	39, 48	jca 554	1 |- ( ph -> ( A e. S /\ A. y e. S y <_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   RR*cxr 10073   <_ cle 10075   - cmin 10266   (,)cioo 12175   [,]cicc 12178   abscabs 13974   ↾_t crest 16081   topGenctg 16098

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  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-tr 4753  df-id 5024  df-eprel 5029  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by:  icccmplem2  22626  icccmplem3  22627