Theorem fin23lem29 9163

Description: Lemma for fin23 9211. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem.a	|- U = seq𝜔 ( ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
fin23lem17.f	|- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
fin23lem.b	|- P = { v e. om | |^| ran U C_ ( t ` v ) }
fin23lem.c	|- Q = ( w e. om |-> ( iota_ x e. P ( x i^i P ) ~~ w ) )
fin23lem.d	|- R = ( w e. om |-> ( iota_ x e. ( om \ P ) ( x i^i ( om \ P ) ) ~~ w ) )
fin23lem.e	|- Z = if ( P e. Fin , ( t o. R ) , ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) )

Assertion

Ref	Expression
fin23lem29	|- U. ran Z C_ U. ran t

Distinct variable groups:   g, i, t, u, v, x, z, a   F, a, t   w, a, x, z, P   v, a, R, i, u   U, a, i, u, v, z   Z, a   g, a

Allowed substitution hints:   P( v, u, t, g, i)   Q( x, z, w, v, u, t, g, i, a)   R( x, z, w, t, g)   U( x, w, t, g)   F( x, z, w, v, u, g, i)   Z( x, z, w, v, u, t, g, i)

Proof of Theorem fin23lem29

Step	Hyp	Ref	Expression
1		fin23lem.e	. 2 |- Z = if ( P e. Fin , ( t o. R ) , ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) )
2		eqif 4126	. . 3 |- ( Z = if ( P e. Fin , ( t o. R ) , ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) ) <-> ( ( P e. Fin /\ Z = ( t o. R ) ) \/ ( -. P e. Fin /\ Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) ) ) )
3	2	biimpi 206	. 2 |- ( Z = if ( P e. Fin , ( t o. R ) , ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) ) -> ( ( P e. Fin /\ Z = ( t o. R ) ) \/ ( -. P e. Fin /\ Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) ) ) )
4		rneq 5351	. . . . . 6 |- ( Z = ( t o. R ) -> ran Z = ran ( t o. R ) )
5	4	unieqd 4446	. . . . 5 |- ( Z = ( t o. R ) -> U. ran Z = U. ran ( t o. R ) )
6		rncoss 5386	. . . . . 6 |- ran ( t o. R ) C_ ran t
7	6	unissi 4461	. . . . 5 |- U. ran ( t o. R ) C_ U. ran t
8	5, 7	syl6eqss 3655	. . . 4 |- ( Z = ( t o. R ) -> U. ran Z C_ U. ran t )
9	8	adantl 482	. . 3 |- ( ( P e. Fin /\ Z = ( t o. R ) ) -> U. ran Z C_ U. ran t )
10		rneq 5351	. . . . . 6 |- ( Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) -> ran Z = ran ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) )
11	10	unieqd 4446	. . . . 5 |- ( Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) -> U. ran Z = U. ran ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) )
12		rncoss 5386	. . . . . . 7 |- ran ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) C_ ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) )
13	12	unissi 4461	. . . . . 6 |- U. ran ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) C_ U. ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) )
14		unissb 4469	. . . . . . 7 |- ( U. ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) C_ U. ran t <-> A. a e. ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) a C_ U. ran t )
15		abid 2610	. . . . . . . . 9 |- ( a e. { a | E. z e. P a = ( ( t ` z ) \ |^| ran U ) } <-> E. z e. P a = ( ( t ` z ) \ |^| ran U ) )
16		fvssunirn 6217	. . . . . . . . . . . . 13 |- ( t ` z ) C_ U. ran t
17	16	a1i 11	. . . . . . . . . . . 12 |- ( z e. P -> ( t ` z ) C_ U. ran t )
18	17	ssdifssd 3748	. . . . . . . . . . 11 |- ( z e. P -> ( ( t ` z ) \ |^| ran U ) C_ U. ran t )
19		sseq1 3626	. . . . . . . . . . 11 |- ( a = ( ( t ` z ) \ |^| ran U ) -> ( a C_ U. ran t <-> ( ( t ` z ) \ |^| ran U ) C_ U. ran t ) )
20	18, 19	syl5ibrcom 237	. . . . . . . . . 10 |- ( z e. P -> ( a = ( ( t ` z ) \ |^| ran U ) -> a C_ U. ran t ) )
21	20	rexlimiv 3027	. . . . . . . . 9 |- ( E. z e. P a = ( ( t ` z ) \ |^| ran U ) -> a C_ U. ran t )
22	15, 21	sylbi 207	. . . . . . . 8 |- ( a e. { a | E. z e. P a = ( ( t ` z ) \ |^| ran U ) } -> a C_ U. ran t )
23		eqid 2622	. . . . . . . . 9 |- ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) = ( z e. P |-> ( ( t ` z ) \ |^| ran U ) )
24	23	rnmpt 5371	. . . . . . . 8 |- ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) = { a | E. z e. P a = ( ( t ` z ) \ |^| ran U ) }
25	22, 24	eleq2s 2719	. . . . . . 7 |- ( a e. ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) -> a C_ U. ran t )
26	14, 25	mprgbir 2927	. . . . . 6 |- U. ran ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) C_ U. ran t
27	13, 26	sstri 3612	. . . . 5 |- U. ran ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) C_ U. ran t
28	11, 27	syl6eqss 3655	. . . 4 |- ( Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) -> U. ran Z C_ U. ran t )
29	28	adantl 482	. . 3 |- ( ( -. P e. Fin /\ Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) ) -> U. ran Z C_ U. ran t )
30	9, 29	jaoi 394	. 2 |- ( ( ( P e. Fin /\ Z = ( t o. R ) ) \/ ( -. P e. Fin /\ Z = ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) ) ) -> U. ran Z C_ U. ran t )
31	1, 3, 30	mp2b 10	1 |- U. ran Z C_ U. ran t

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   o. ccom 5118   suc csuc 5725   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   omcom 7065  seq_𝜔cseqom 7542   ^m cmap 7857   ~~ cen 7952   Fincfn 7955

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-opab 4713  df-mpt 4730  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896

This theorem is referenced by:  fin23lem31  9165