Theorem fin23lem33 9167

Description: Lemma for fin23 9211. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem33.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 ) }

Assertion

Ref	Expression
fin23lem33	|- ( G e. F -> E. f A. b ( ( b : om -1-1-> _V /\ U. ran b C_ G ) -> ( ( f ` b ) : om -1-1-> _V /\ U. ran ( f ` b ) C. U. ran b ) ) )

Distinct variable groups:   a, b, f, g, x, G   F, a

Allowed substitution hints:   F( x, f, g, b)

Proof of Theorem fin23lem33

Dummy variables c d e i j k l y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( j = c -> ( e ` j ) = ( e ` c ) )
2	1	ineq1d 3813	. . . . . 6 |- ( j = c -> ( ( e ` j ) i^i k ) = ( ( e ` c ) i^i k ) )
3	2	eqeq1d 2624	. . . . 5 |- ( j = c -> ( ( ( e ` j ) i^i k ) = (/) <-> ( ( e ` c ) i^i k ) = (/) ) )
4	3, 2	ifbieq2d 4111	. . . 4 |- ( j = c -> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) = if ( ( ( e ` c ) i^i k ) = (/) , k , ( ( e ` c ) i^i k ) ) )
5		ineq2 3808	. . . . . 6 |- ( k = d -> ( ( e ` c ) i^i k ) = ( ( e ` c ) i^i d ) )
6	5	eqeq1d 2624	. . . . 5 |- ( k = d -> ( ( ( e ` c ) i^i k ) = (/) <-> ( ( e ` c ) i^i d ) = (/) ) )
7		id 22	. . . . 5 |- ( k = d -> k = d )
8	6, 7, 5	ifbieq12d 4113	. . . 4 |- ( k = d -> if ( ( ( e ` c ) i^i k ) = (/) , k , ( ( e ` c ) i^i k ) ) = if ( ( ( e ` c ) i^i d ) = (/) , d , ( ( e ` c ) i^i d ) ) )
9	4, 8	cbvmpt2v 6735	. . 3 |- ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) = ( c e. om , d e. _V |-> if ( ( ( e ` c ) i^i d ) = (/) , d , ( ( e ` c ) i^i d ) ) )
10		eqid 2622	. . 3 |- U. ran e = U. ran e
11		seqomeq12 7549	. . 3 |- ( ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) = ( c e. om , d e. _V |-> if ( ( ( e ` c ) i^i d ) = (/) , d , ( ( e ` c ) i^i d ) ) ) /\ U. ran e = U. ran e ) -> seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) = seq𝜔 ( ( c e. om , d e. _V |-> if ( ( ( e ` c ) i^i d ) = (/) , d , ( ( e ` c ) i^i d ) ) ) , U. ran e ) )
12	9, 10, 11	mp2an 708	. 2 |- seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) = seq𝜔 ( ( c e. om , d e. _V |-> if ( ( ( e ` c ) i^i d ) = (/) , d , ( ( e ` c ) i^i d ) ) ) , U. ran e )
13		fin23lem33.f	. 2 |- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
14		fveq2 6191	. . . 4 |- ( l = y -> ( e ` l ) = ( e ` y ) )
15	14	sseq2d 3633	. . 3 |- ( l = y -> ( |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) <-> |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` y ) ) )
16	15	cbvrabv 3199	. 2 |- { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } = { y e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` y ) }
17		eqid 2622	. 2 |- ( g e. om |-> ( iota_ x e. { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ( x i^i { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ~~ g ) ) = ( g e. om |-> ( iota_ x e. { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ( x i^i { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ~~ g ) )
18		eqid 2622	. 2 |- ( g e. om |-> ( iota_ x e. ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ( x i^i ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ) ~~ g ) ) = ( g e. om |-> ( iota_ x e. ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ( x i^i ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ) ~~ g ) )
19		eqid 2622	. 2 |- if ( { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } e. Fin , ( e o. ( g e. om |-> ( iota_ x e. ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ( x i^i ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ) ~~ g ) ) ) , ( ( i e. { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } |-> ( ( e ` i ) \ |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) ) ) o. ( g e. om |-> ( iota_ x e. { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ( x i^i { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ~~ g ) ) ) ) = if ( { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } e. Fin , ( e o. ( g e. om |-> ( iota_ x e. ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ( x i^i ( om \ { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ) ~~ g ) ) ) , ( ( i e. { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } |-> ( ( e ` i ) \ |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) ) ) o. ( g e. om |-> ( iota_ x e. { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ( x i^i { l e. om | |^| ran seq𝜔 ( ( j e. om , k e. _V |-> if ( ( ( e ` j ) i^i k ) = (/) , k , ( ( e ` j ) i^i k ) ) ) , U. ran e ) C_ ( e ` l ) } ) ~~ g ) ) ) )
20	12, 13, 16, 17, 18, 19	fin23lem32 9166	1 |- ( G e. F -> E. f A. b ( ( b : om -1-1-> _V /\ U. ran b C_ G ) -> ( ( f ` b ) : om -1-1-> _V /\ U. ran ( f ` b ) C. U. ran b ) ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)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   -1-1->wf1 5885   ` 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-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-se 5074  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-pred 5680  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin23lem41  9174