Theorem fin23lem21 9161

Description: Lemma for fin23 9211. X is not empty. We only need here that t has at least one set in its range besides (/); the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-May-2015.)

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 ) }

Assertion

Ref	Expression
fin23lem21	|- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U =/= (/) )

Distinct variable groups:   g, i, t, u, x, a   F, a, t   V, a   x, a   U, a, i, u   g, a

Allowed substitution hints:   U( x, t, g)   F( x, u, g, i)   V( x, u, t, g, i)

Proof of Theorem fin23lem21

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		fin23lem17.f	. . 3 |- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
3	1, 2	fin23lem17 9160	. 2 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U e. ran U )
4	1	fnseqom 7550	. . . . 5 |- U Fn om
5		fvelrnb 6243	. . . . 5 |- ( U Fn om -> ( |^| ran U e. ran U <-> E. a e. om ( U ` a ) = |^| ran U ) )
6	4, 5	ax-mp 5	. . . 4 |- ( |^| ran U e. ran U <-> E. a e. om ( U ` a ) = |^| ran U )
7		id 22	. . . . . . 7 |- ( a e. om -> a e. om )
8		vex 3203	. . . . . . . . . 10 |- t e. _V
9		f1f1orn 6148	. . . . . . . . . 10 |- ( t : om -1-1-> V -> t : om -1-1-onto-> ran t )
10		f1oen3g 7971	. . . . . . . . . 10 |- ( ( t e. _V /\ t : om -1-1-onto-> ran t ) -> om ~~ ran t )
11	8, 9, 10	sylancr 695	. . . . . . . . 9 |- ( t : om -1-1-> V -> om ~~ ran t )
12		ominf 8172	. . . . . . . . 9 |- -. om e. Fin
13		ssdif0 3942	. . . . . . . . . . 11 |- ( ran t C_ { (/) } <-> ( ran t \ { (/) } ) = (/) )
14		snfi 8038	. . . . . . . . . . . . 13 |- { (/) } e. Fin
15		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( { (/) } e. Fin /\ ran t C_ { (/) } ) -> ran t e. Fin )
16	14, 15	mpan 706	. . . . . . . . . . . 12 |- ( ran t C_ { (/) } -> ran t e. Fin )
17		enfi 8176	. . . . . . . . . . . 12 |- ( om ~~ ran t -> ( om e. Fin <-> ran t e. Fin ) )
18	16, 17	syl5ibr 236	. . . . . . . . . . 11 |- ( om ~~ ran t -> ( ran t C_ { (/) } -> om e. Fin ) )
19	13, 18	syl5bir 233	. . . . . . . . . 10 |- ( om ~~ ran t -> ( ( ran t \ { (/) } ) = (/) -> om e. Fin ) )
20	19	necon3bd 2808	. . . . . . . . 9 |- ( om ~~ ran t -> ( -. om e. Fin -> ( ran t \ { (/) } ) =/= (/) ) )
21	11, 12, 20	mpisyl 21	. . . . . . . 8 |- ( t : om -1-1-> V -> ( ran t \ { (/) } ) =/= (/) )
22		n0 3931	. . . . . . . . 9 |- ( ( ran t \ { (/) } ) =/= (/) <-> E. a a e. ( ran t \ { (/) } ) )
23		eldifsn 4317	. . . . . . . . . . 11 |- ( a e. ( ran t \ { (/) } ) <-> ( a e. ran t /\ a =/= (/) ) )
24		elssuni 4467	. . . . . . . . . . . 12 |- ( a e. ran t -> a C_ U. ran t )
25		ssn0 3976	. . . . . . . . . . . 12 |- ( ( a C_ U. ran t /\ a =/= (/) ) -> U. ran t =/= (/) )
26	24, 25	sylan 488	. . . . . . . . . . 11 |- ( ( a e. ran t /\ a =/= (/) ) -> U. ran t =/= (/) )
27	23, 26	sylbi 207	. . . . . . . . . 10 |- ( a e. ( ran t \ { (/) } ) -> U. ran t =/= (/) )
28	27	exlimiv 1858	. . . . . . . . 9 |- ( E. a a e. ( ran t \ { (/) } ) -> U. ran t =/= (/) )
29	22, 28	sylbi 207	. . . . . . . 8 |- ( ( ran t \ { (/) } ) =/= (/) -> U. ran t =/= (/) )
30	21, 29	syl 17	. . . . . . 7 |- ( t : om -1-1-> V -> U. ran t =/= (/) )
31	1	fin23lem14 9155	. . . . . . 7 |- ( ( a e. om /\ U. ran t =/= (/) ) -> ( U ` a ) =/= (/) )
32	7, 30, 31	syl2anr 495	. . . . . 6 |- ( ( t : om -1-1-> V /\ a e. om ) -> ( U ` a ) =/= (/) )
33		neeq1 2856	. . . . . 6 |- ( ( U ` a ) = |^| ran U -> ( ( U ` a ) =/= (/) <-> |^| ran U =/= (/) ) )
34	32, 33	syl5ibcom 235	. . . . 5 |- ( ( t : om -1-1-> V /\ a e. om ) -> ( ( U ` a ) = |^| ran U -> |^| ran U =/= (/) ) )
35	34	rexlimdva 3031	. . . 4 |- ( t : om -1-1-> V -> ( E. a e. om ( U ` a ) = |^| ran U -> |^| ran U =/= (/) ) )
36	6, 35	syl5bi 232	. . 3 |- ( t : om -1-1-> V -> ( |^| ran U e. ran U -> |^| ran U =/= (/) ) )
37	36	adantl 482	. 2 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> ( |^| ran U e. ran U -> |^| ran U =/= (/) ) )
38	3, 37	mpd 15	1 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   |^|cint 4475   class class class wbr 4653   ran crn 5115   suc csuc 5725   Fn wfn 5883   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (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  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-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-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  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  fin23lem31  9165