Theorem fin23lem16 9157

Description: Lemma for fin23 9211. U ranges over the original set; in particular ran U is a set, although we do not assume here that U is. (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
fin23lem16	|- U. ran U = U. ran t

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

Allowed substitution hint:   U( t)

Proof of Theorem fin23lem16

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

Step	Hyp	Ref	Expression
1		unissb 4469	. . 3 |- ( U. ran U C_ U. ran t <-> A. a e. ran U a C_ U. ran t )
2		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 )
3	2	fnseqom 7550	. . . . 5 |- U Fn om
4		fvelrnb 6243	. . . . 5 |- ( U Fn om -> ( a e. ran U <-> E. b e. om ( U ` b ) = a ) )
5	3, 4	ax-mp 5	. . . 4 |- ( a e. ran U <-> E. b e. om ( U ` b ) = a )
6		peano1 7085	. . . . . . . 8 |- (/) e. om
7		0ss 3972	. . . . . . . . 9 |- (/) C_ b
8	2	fin23lem15 9156	. . . . . . . . 9 |- ( ( ( b e. om /\ (/) e. om ) /\ (/) C_ b ) -> ( U ` b ) C_ ( U ` (/) ) )
9	7, 8	mpan2 707	. . . . . . . 8 |- ( ( b e. om /\ (/) e. om ) -> ( U ` b ) C_ ( U ` (/) ) )
10	6, 9	mpan2 707	. . . . . . 7 |- ( b e. om -> ( U ` b ) C_ ( U ` (/) ) )
11		vex 3203	. . . . . . . . . 10 |- t e. _V
12	11	rnex 7100	. . . . . . . . 9 |- ran t e. _V
13	12	uniex 6953	. . . . . . . 8 |- U. ran t e. _V
14	2	seqom0g 7551	. . . . . . . 8 |- ( U. ran t e. _V -> ( U ` (/) ) = U. ran t )
15	13, 14	ax-mp 5	. . . . . . 7 |- ( U ` (/) ) = U. ran t
16	10, 15	syl6sseq 3651	. . . . . 6 |- ( b e. om -> ( U ` b ) C_ U. ran t )
17		sseq1 3626	. . . . . 6 |- ( ( U ` b ) = a -> ( ( U ` b ) C_ U. ran t <-> a C_ U. ran t ) )
18	16, 17	syl5ibcom 235	. . . . 5 |- ( b e. om -> ( ( U ` b ) = a -> a C_ U. ran t ) )
19	18	rexlimiv 3027	. . . 4 |- ( E. b e. om ( U ` b ) = a -> a C_ U. ran t )
20	5, 19	sylbi 207	. . 3 |- ( a e. ran U -> a C_ U. ran t )
21	1, 20	mprgbir 2927	. 2 |- U. ran U C_ U. ran t
22		fnfvelrn 6356	. . . . 5 |- ( ( U Fn om /\ (/) e. om ) -> ( U ` (/) ) e. ran U )
23	3, 6, 22	mp2an 708	. . . 4 |- ( U ` (/) ) e. ran U
24	15, 23	eqeltrri 2698	. . 3 |- U. ran t e. ran U
25		elssuni 4467	. . 3 |- ( U. ran t e. ran U -> U. ran t C_ U. ran U )
26	24, 25	ax-mp 5	. 2 |- U. ran t C_ U. ran U
27	21, 26	eqssi 3619	1 |- U. ran U = U. ran t

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   U.cuni 4436   ran crn 5115   Fn wfn 5883   ` 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:  fin23lem17  9160  fin23lem31  9165