Theorem fin23lem12 9153

Description: The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of U and its intersection. First, the value of U at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

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 )

Assertion

Ref	Expression
fin23lem12	|- ( A e. om -> ( U ` suc A ) = if ( ( ( t ` A ) i^i ( U ` A ) ) = (/) , ( U ` A ) , ( ( t ` A ) i^i ( U ` A ) ) ) )

Distinct variable groups:   t, i, u   A, i, u   U, i, u

Allowed substitution hints:   A( t)   U( t)

Proof of Theorem fin23lem12

Step	Hyp	Ref	Expression
1		fin23lem.a	. . 3 |- U = seq𝜔 ( ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
2	1	seqomsuc 7552	. 2 |- ( A e. om -> ( U ` suc A ) = ( A ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) ( U ` A ) ) )
3		fvex 6201	. . 3 |- ( U ` A ) e. _V
4		fveq2 6191	. . . . . . 7 |- ( i = A -> ( t ` i ) = ( t ` A ) )
5	4	ineq1d 3813	. . . . . 6 |- ( i = A -> ( ( t ` i ) i^i u ) = ( ( t ` A ) i^i u ) )
6	5	eqeq1d 2624	. . . . 5 |- ( i = A -> ( ( ( t ` i ) i^i u ) = (/) <-> ( ( t ` A ) i^i u ) = (/) ) )
7	6, 5	ifbieq2d 4111	. . . 4 |- ( i = A -> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) = if ( ( ( t ` A ) i^i u ) = (/) , u , ( ( t ` A ) i^i u ) ) )
8		ineq2 3808	. . . . . 6 |- ( u = ( U ` A ) -> ( ( t ` A ) i^i u ) = ( ( t ` A ) i^i ( U ` A ) ) )
9	8	eqeq1d 2624	. . . . 5 |- ( u = ( U ` A ) -> ( ( ( t ` A ) i^i u ) = (/) <-> ( ( t ` A ) i^i ( U ` A ) ) = (/) ) )
10		id 22	. . . . 5 |- ( u = ( U ` A ) -> u = ( U ` A ) )
11	9, 10, 8	ifbieq12d 4113	. . . 4 |- ( u = ( U ` A ) -> if ( ( ( t ` A ) i^i u ) = (/) , u , ( ( t ` A ) i^i u ) ) = if ( ( ( t ` A ) i^i ( U ` A ) ) = (/) , ( U ` A ) , ( ( t ` A ) i^i ( U ` A ) ) ) )
12		eqid 2622	. . . 4 |- ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) = ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) )
13	3	inex2 4800	. . . . 5 |- ( ( t ` A ) i^i ( U ` A ) ) e. _V
14	3, 13	ifex 4156	. . . 4 |- if ( ( ( t ` A ) i^i ( U ` A ) ) = (/) , ( U ` A ) , ( ( t ` A ) i^i ( U ` A ) ) ) e. _V
15	7, 11, 12, 14	ovmpt2 6796	. . 3 |- ( ( A e. om /\ ( U ` A ) e. _V ) -> ( A ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) ( U ` A ) ) = if ( ( ( t ` A ) i^i ( U ` A ) ) = (/) , ( U ` A ) , ( ( t ` A ) i^i ( U ` A ) ) ) )
16	3, 15	mpan2 707	. 2 |- ( A e. om -> ( A ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) ( U ` A ) ) = if ( ( ( t ` A ) i^i ( U ` A ) ) = (/) , ( U ` A ) , ( ( t ` A ) i^i ( U ` A ) ) ) )
17	2, 16	eqtrd 2656	1 |- ( A e. om -> ( U ` suc A ) = if ( ( ( t ` A ) i^i ( U ` A ) ) = (/) , ( U ` A ) , ( ( t ` A ) i^i ( U ` A ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ifcif 4086   U.cuni 4436   ran crn 5115   suc csuc 5725   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065  seq_𝜔cseqom 7542

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

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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543

This theorem is referenced by:  fin23lem13  9154  fin23lem14  9155  fin23lem19  9158