Theorem fin23lem17 9160

Description: Lemma for fin23 9211. By ? Fin3DS ? , U achieves its minimum ( X in the synopsis above, but we will not be assigning a symbol here). TODO: Fix comment; math symbol Fin3DS does not exist. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-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
fin23lem17	|- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U e. 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 fin23lem17

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

Step	Hyp	Ref	Expression
1		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 )
2	1	fnseqom 7550	. . . . 5 |- U Fn om
3		dffn3 6054	. . . . 5 |- ( U Fn om <-> U : om --> ran U )
4	2, 3	mpbi 220	. . . 4 |- U : om --> ran U
5		pwuni 4474	. . . . 5 |- ran U C_ ~P U. ran U
6	1	fin23lem16 9157	. . . . . 6 |- U. ran U = U. ran t
7	6	pweqi 4162	. . . . 5 |- ~P U. ran U = ~P U. ran t
8	5, 7	sseqtri 3637	. . . 4 |- ran U C_ ~P U. ran t
9		fss 6056	. . . 4 |- ( ( U : om --> ran U /\ ran U C_ ~P U. ran t ) -> U : om --> ~P U. ran t )
10	4, 8, 9	mp2an 708	. . 3 |- U : om --> ~P U. ran t
11		vex 3203	. . . . . . 7 |- t e. _V
12	11	rnex 7100	. . . . . 6 |- ran t e. _V
13	12	uniex 6953	. . . . 5 |- U. ran t e. _V
14	13	pwex 4848	. . . 4 |- ~P U. ran t e. _V
15		f1f 6101	. . . . . 6 |- ( t : om -1-1-> V -> t : om --> V )
16		dmfex 7124	. . . . . 6 |- ( ( t e. _V /\ t : om --> V ) -> om e. _V )
17	11, 15, 16	sylancr 695	. . . . 5 |- ( t : om -1-1-> V -> om e. _V )
18	17	adantl 482	. . . 4 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> om e. _V )
19		elmapg 7870	. . . 4 |- ( ( ~P U. ran t e. _V /\ om e. _V ) -> ( U e. ( ~P U. ran t ^m om ) <-> U : om --> ~P U. ran t ) )
20	14, 18, 19	sylancr 695	. . 3 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> ( U e. ( ~P U. ran t ^m om ) <-> U : om --> ~P U. ran t ) )
21	10, 20	mpbiri 248	. 2 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> U e. ( ~P U. ran t ^m om ) )
22		fin23lem17.f	. . . . 5 |- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
23	22	isfin3ds 9151	. . . 4 |- ( U. ran t e. F -> ( U. ran t e. F <-> A. b e. ( ~P U. ran t ^m om ) ( A. c e. om ( b ` suc c ) C_ ( b ` c ) -> |^| ran b e. ran b ) ) )
24	23	ibi 256	. . 3 |- ( U. ran t e. F -> A. b e. ( ~P U. ran t ^m om ) ( A. c e. om ( b ` suc c ) C_ ( b ` c ) -> |^| ran b e. ran b ) )
25	24	adantr 481	. 2 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> A. b e. ( ~P U. ran t ^m om ) ( A. c e. om ( b ` suc c ) C_ ( b ` c ) -> |^| ran b e. ran b ) )
26	1	fin23lem13 9154	. . . 4 |- ( c e. om -> ( U ` suc c ) C_ ( U ` c ) )
27	26	rgen 2922	. . 3 |- A. c e. om ( U ` suc c ) C_ ( U ` c )
28	27	a1i 11	. 2 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> A. c e. om ( U ` suc c ) C_ ( U ` c ) )
29		fveq1 6190	. . . . . 6 |- ( b = U -> ( b ` suc c ) = ( U ` suc c ) )
30		fveq1 6190	. . . . . 6 |- ( b = U -> ( b ` c ) = ( U ` c ) )
31	29, 30	sseq12d 3634	. . . . 5 |- ( b = U -> ( ( b ` suc c ) C_ ( b ` c ) <-> ( U ` suc c ) C_ ( U ` c ) ) )
32	31	ralbidv 2986	. . . 4 |- ( b = U -> ( A. c e. om ( b ` suc c ) C_ ( b ` c ) <-> A. c e. om ( U ` suc c ) C_ ( U ` c ) ) )
33		rneq 5351	. . . . . 6 |- ( b = U -> ran b = ran U )
34	33	inteqd 4480	. . . . 5 |- ( b = U -> |^| ran b = |^| ran U )
35	34, 33	eleq12d 2695	. . . 4 |- ( b = U -> ( |^| ran b e. ran b <-> |^| ran U e. ran U ) )
36	32, 35	imbi12d 334	. . 3 |- ( b = U -> ( ( A. c e. om ( b ` suc c ) C_ ( b ` c ) -> |^| ran b e. ran b ) <-> ( A. c e. om ( U ` suc c ) C_ ( U ` c ) -> |^| ran U e. ran U ) ) )
37	36	rspcv 3305	. 2 |- ( U e. ( ~P U. ran t ^m om ) -> ( A. b e. ( ~P U. ran t ^m om ) ( A. c e. om ( b ` suc c ) C_ ( b ` c ) -> |^| ran b e. ran b ) -> ( A. c e. om ( U ` suc c ) C_ ( U ` c ) -> |^| ran U e. ran U ) ) )
38	21, 25, 28, 37	syl3c 66	1 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U e. ran U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   ran crn 5115   suc csuc 5725   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065  seq_𝜔cseqom 7542   ^m cmap 7857

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-map 7859

This theorem is referenced by:  fin23lem21  9161