Theorem fin23lem14 9155

Description: Lemma for fin23 9211. U will never evolve to an empty set if it did not start with one. (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
fin23lem14	|- ( ( A e. om /\ U. ran t =/= (/) ) -> ( U ` A ) =/= (/) )

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

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

Proof of Theorem fin23lem14

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( a = (/) -> ( U ` a ) = ( U ` (/) ) )
2	1	neeq1d 2853	. . . 4 |- ( a = (/) -> ( ( U ` a ) =/= (/) <-> ( U ` (/) ) =/= (/) ) )
3	2	imbi2d 330	. . 3 |- ( a = (/) -> ( ( U. ran t =/= (/) -> ( U ` a ) =/= (/) ) <-> ( U. ran t =/= (/) -> ( U ` (/) ) =/= (/) ) ) )
4		fveq2 6191	. . . . 5 |- ( a = b -> ( U ` a ) = ( U ` b ) )
5	4	neeq1d 2853	. . . 4 |- ( a = b -> ( ( U ` a ) =/= (/) <-> ( U ` b ) =/= (/) ) )
6	5	imbi2d 330	. . 3 |- ( a = b -> ( ( U. ran t =/= (/) -> ( U ` a ) =/= (/) ) <-> ( U. ran t =/= (/) -> ( U ` b ) =/= (/) ) ) )
7		fveq2 6191	. . . . 5 |- ( a = suc b -> ( U ` a ) = ( U ` suc b ) )
8	7	neeq1d 2853	. . . 4 |- ( a = suc b -> ( ( U ` a ) =/= (/) <-> ( U ` suc b ) =/= (/) ) )
9	8	imbi2d 330	. . 3 |- ( a = suc b -> ( ( U. ran t =/= (/) -> ( U ` a ) =/= (/) ) <-> ( U. ran t =/= (/) -> ( U ` suc b ) =/= (/) ) ) )
10		fveq2 6191	. . . . 5 |- ( a = A -> ( U ` a ) = ( U ` A ) )
11	10	neeq1d 2853	. . . 4 |- ( a = A -> ( ( U ` a ) =/= (/) <-> ( U ` A ) =/= (/) ) )
12	11	imbi2d 330	. . 3 |- ( a = A -> ( ( U. ran t =/= (/) -> ( U ` a ) =/= (/) ) <-> ( U. ran t =/= (/) -> ( U ` A ) =/= (/) ) ) )
13		vex 3203	. . . . . . 7 |- t e. _V
14	13	rnex 7100	. . . . . 6 |- ran t e. _V
15	14	uniex 6953	. . . . 5 |- U. ran t e. _V
16		fin23lem.a	. . . . . 6 |- U = seq𝜔 ( ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
17	16	seqom0g 7551	. . . . 5 |- ( U. ran t e. _V -> ( U ` (/) ) = U. ran t )
18	15, 17	mp1i 13	. . . 4 |- ( U. ran t =/= (/) -> ( U ` (/) ) = U. ran t )
19		id 22	. . . 4 |- ( U. ran t =/= (/) -> U. ran t =/= (/) )
20	18, 19	eqnetrd 2861	. . 3 |- ( U. ran t =/= (/) -> ( U ` (/) ) =/= (/) )
21	16	fin23lem12 9153	. . . . . . 7 |- ( b e. om -> ( U ` suc b ) = if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) )
22	21	adantr 481	. . . . . 6 |- ( ( b e. om /\ ( U ` b ) =/= (/) ) -> ( U ` suc b ) = if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) )
23		iftrue 4092	. . . . . . . . 9 |- ( ( ( t ` b ) i^i ( U ` b ) ) = (/) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) = ( U ` b ) )
24	23	adantr 481	. . . . . . . 8 |- ( ( ( ( t ` b ) i^i ( U ` b ) ) = (/) /\ ( b e. om /\ ( U ` b ) =/= (/) ) ) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) = ( U ` b ) )
25		simprr 796	. . . . . . . 8 |- ( ( ( ( t ` b ) i^i ( U ` b ) ) = (/) /\ ( b e. om /\ ( U ` b ) =/= (/) ) ) -> ( U ` b ) =/= (/) )
26	24, 25	eqnetrd 2861	. . . . . . 7 |- ( ( ( ( t ` b ) i^i ( U ` b ) ) = (/) /\ ( b e. om /\ ( U ` b ) =/= (/) ) ) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) =/= (/) )
27		iffalse 4095	. . . . . . . . 9 |- ( -. ( ( t ` b ) i^i ( U ` b ) ) = (/) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) = ( ( t ` b ) i^i ( U ` b ) ) )
28	27	adantr 481	. . . . . . . 8 |- ( ( -. ( ( t ` b ) i^i ( U ` b ) ) = (/) /\ ( b e. om /\ ( U ` b ) =/= (/) ) ) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) = ( ( t ` b ) i^i ( U ` b ) ) )
29		df-ne 2795	. . . . . . . . . 10 |- ( ( ( t ` b ) i^i ( U ` b ) ) =/= (/) <-> -. ( ( t ` b ) i^i ( U ` b ) ) = (/) )
30	29	biimpri 218	. . . . . . . . 9 |- ( -. ( ( t ` b ) i^i ( U ` b ) ) = (/) -> ( ( t ` b ) i^i ( U ` b ) ) =/= (/) )
31	30	adantr 481	. . . . . . . 8 |- ( ( -. ( ( t ` b ) i^i ( U ` b ) ) = (/) /\ ( b e. om /\ ( U ` b ) =/= (/) ) ) -> ( ( t ` b ) i^i ( U ` b ) ) =/= (/) )
32	28, 31	eqnetrd 2861	. . . . . . 7 |- ( ( -. ( ( t ` b ) i^i ( U ` b ) ) = (/) /\ ( b e. om /\ ( U ` b ) =/= (/) ) ) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) =/= (/) )
33	26, 32	pm2.61ian 831	. . . . . 6 |- ( ( b e. om /\ ( U ` b ) =/= (/) ) -> if ( ( ( t ` b ) i^i ( U ` b ) ) = (/) , ( U ` b ) , ( ( t ` b ) i^i ( U ` b ) ) ) =/= (/) )
34	22, 33	eqnetrd 2861	. . . . 5 |- ( ( b e. om /\ ( U ` b ) =/= (/) ) -> ( U ` suc b ) =/= (/) )
35	34	ex 450	. . . 4 |- ( b e. om -> ( ( U ` b ) =/= (/) -> ( U ` suc b ) =/= (/) ) )
36	35	imim2d 57	. . 3 |- ( b e. om -> ( ( U. ran t =/= (/) -> ( U ` b ) =/= (/) ) -> ( U. ran t =/= (/) -> ( U ` suc b ) =/= (/) ) ) )
37	3, 6, 9, 12, 20, 36	finds 7092	. 2 |- ( A e. om -> ( U. ran t =/= (/) -> ( U ` A ) =/= (/) ) )
38	37	imp 445	1 |- ( ( A e. om /\ U. ran t =/= (/) ) -> ( U ` A ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ifcif 4086   U.cuni 4436   ran crn 5115   suc csuc 5725   ` cfv 5888   |-> 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:  fin23lem21  9161