Theorem noetalem2 31864

Description: Lemma for noeta 31868. Z is an upper bound for A. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 4-Dec-2021.)

Hypotheses

Ref	Expression
noetalem.1	|- S = if ( E. x e. A A. y e. A -. x <s y , ( ( iota_ x e. A A. y e. A -. x <s y ) u. { <. dom ( iota_ x e. A A. y e. A -. x <s y ) , 2o >. } ) , ( g e. { y | E. u e. A ( y e. dom u /\ A. v e. A ( -. v <s u -> ( u |` suc y ) = ( v |` suc y ) ) ) } |-> ( iota x E. u e. A ( g e. dom u /\ A. v e. A ( -. v <s u -> ( u |` suc g ) = ( v |` suc g ) ) /\ ( u ` g ) = x ) ) ) )
noetalem.2	|- Z = ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) )

Assertion

Ref	Expression
noetalem2	|- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> X <s Z )

Distinct variable groups:   A, g, u, v, x, y   u, X, v, x, y

Allowed substitution hints:   B( x, y, v, u, g)   S( x, y, v, u, g)   X( g)   Z( x, y, v, u, g)

Proof of Theorem noetalem2

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . 4 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> A C_ No )
2		simpl2 1065	. . . 4 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> A e. _V )
3		simpr 477	. . . 4 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> X e. A )
4		noetalem.1	. . . . 5 |- S = if ( E. x e. A A. y e. A -. x <s y , ( ( iota_ x e. A A. y e. A -. x <s y ) u. { <. dom ( iota_ x e. A A. y e. A -. x <s y ) , 2o >. } ) , ( g e. { y | E. u e. A ( y e. dom u /\ A. v e. A ( -. v <s u -> ( u |` suc y ) = ( v |` suc y ) ) ) } |-> ( iota x E. u e. A ( g e. dom u /\ A. v e. A ( -. v <s u -> ( u |` suc g ) = ( v |` suc g ) ) /\ ( u ` g ) = x ) ) ) )
5	4	nosupbnd1 31860	. . . 4 |- ( ( A C_ No /\ A e. _V /\ X e. A ) -> ( X |` dom S ) <s S )
6	1, 2, 3, 5	syl3anc 1326	. . 3 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> ( X |` dom S ) <s S )
7		noetalem.2	. . . . . 6 |- Z = ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) )
8	7	reseq1i 5392	. . . . 5 |- ( Z |` dom S ) = ( ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) |` dom S )
9		resundir 5411	. . . . . 6 |- ( ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) |` dom S ) = ( ( S |` dom S ) u. ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) |` dom S ) )
10		df-res 5126	. . . . . . . 8 |- ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) |` dom S ) = ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) i^i ( dom S X. _V ) )
11		incom 3805	. . . . . . . . . 10 |- ( ( suc U. ( bday " B ) \ dom S ) i^i dom S ) = ( dom S i^i ( suc U. ( bday " B ) \ dom S ) )
12		disjdif 4040	. . . . . . . . . 10 |- ( dom S i^i ( suc U. ( bday " B ) \ dom S ) ) = (/)
13	11, 12	eqtri 2644	. . . . . . . . 9 |- ( ( suc U. ( bday " B ) \ dom S ) i^i dom S ) = (/)
14		xpdisj1 5555	. . . . . . . . 9 |- ( ( ( suc U. ( bday " B ) \ dom S ) i^i dom S ) = (/) -> ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) i^i ( dom S X. _V ) ) = (/) )
15	13, 14	ax-mp 5	. . . . . . . 8 |- ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) i^i ( dom S X. _V ) ) = (/)
16	10, 15	eqtri 2644	. . . . . . 7 |- ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) |` dom S ) = (/)
17	16	uneq2i 3764	. . . . . 6 |- ( ( S |` dom S ) u. ( ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) |` dom S ) ) = ( ( S |` dom S ) u. (/) )
18		un0 3967	. . . . . 6 |- ( ( S |` dom S ) u. (/) ) = ( S |` dom S )
19	9, 17, 18	3eqtri 2648	. . . . 5 |- ( ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) |` dom S ) = ( S |` dom S )
20	8, 19	eqtri 2644	. . . 4 |- ( Z |` dom S ) = ( S |` dom S )
21	4	nosupno 31849	. . . . . . 7 |- ( ( A C_ No /\ A e. _V ) -> S e. No )
22	1, 2, 21	syl2anc 693	. . . . . 6 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> S e. No )
23		nofun 31802	. . . . . 6 |- ( S e. No -> Fun S )
24	22, 23	syl 17	. . . . 5 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> Fun S )
25		funrel 5905	. . . . 5 |- ( Fun S -> Rel S )
26		resdm 5441	. . . . 5 |- ( Rel S -> ( S |` dom S ) = S )
27	24, 25, 26	3syl 18	. . . 4 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> ( S |` dom S ) = S )
28	20, 27	syl5eq 2668	. . 3 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> ( Z |` dom S ) = S )
29	6, 28	breqtrrd 4681	. 2 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> ( X |` dom S ) <s ( Z |` dom S ) )
30		simp1 1061	. . . 4 |- ( ( A C_ No /\ A e. _V /\ B e. _V ) -> A C_ No )
31	30	sselda 3603	. . 3 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> X e. No )
32	4, 7	noetalem1 31863	. . . 4 |- ( ( A C_ No /\ A e. _V /\ B e. _V ) -> Z e. No )
33	32	adantr 481	. . 3 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> Z e. No )
34		nodmon 31803	. . . 4 |- ( S e. No -> dom S e. On )
35	22, 34	syl 17	. . 3 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> dom S e. On )
36		sltres 31815	. . 3 |- ( ( X e. No /\ Z e. No /\ dom S e. On ) -> ( ( X |` dom S ) <s ( Z |` dom S ) -> X <s Z ) )
37	31, 33, 35, 36	syl3anc 1326	. 2 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> ( ( X |` dom S ) <s ( Z |` dom S ) -> X <s Z ) )
38	29, 37	mpd 15	1 |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> X <s Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   "cima 5117   Rel wrel 5119   Oncon0 5723   suc csuc 5725   iotacio 5849   Fun wfun 5882   ` cfv 5888   iota_crio 6610   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794   bdaycbday 31795

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-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-int 4476  df-iun 4522  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-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-riota 6611  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798

This theorem is referenced by:  noetalem5  31867