Theorem fseqen 8850

Description: A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
fseqen	|- ( ( ( A X. A ) ~~ A /\ A =/= (/) ) -> U_ n e. om ( A ^m n ) ~~ ( om X. A ) )

Distinct variable group:   A, n

Proof of Theorem fseqen

Dummy variables f b g k x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 7964	. 2 |- ( ( A X. A ) ~~ A <-> E. f f : ( A X. A ) -1-1-onto-> A )
2		n0 3931	. 2 |- ( A =/= (/) <-> E. b b e. A )
3		eeanv 2182	. . 3 |- ( E. f E. b ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) <-> ( E. f f : ( A X. A ) -1-1-onto-> A /\ E. b b e. A ) )
4		omex 8540	. . . . . . 7 |- om e. _V
5		simpl 473	. . . . . . . . 9 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> f : ( A X. A ) -1-1-onto-> A )
6		f1ofo 6144	. . . . . . . . 9 |- ( f : ( A X. A ) -1-1-onto-> A -> f : ( A X. A ) -onto-> A )
7		forn 6118	. . . . . . . . 9 |- ( f : ( A X. A ) -onto-> A -> ran f = A )
8	5, 6, 7	3syl 18	. . . . . . . 8 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> ran f = A )
9		vex 3203	. . . . . . . . 9 |- f e. _V
10	9	rnex 7100	. . . . . . . 8 |- ran f e. _V
11	8, 10	syl6eqelr 2710	. . . . . . 7 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> A e. _V )
12		xpexg 6960	. . . . . . 7 |- ( ( om e. _V /\ A e. _V ) -> ( om X. A ) e. _V )
13	4, 11, 12	sylancr 695	. . . . . 6 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> ( om X. A ) e. _V )
14		simpr 477	. . . . . . 7 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> b e. A )
15		eqid 2622	. . . . . . 7 |- seq𝜔 ( ( k e. _V , g e. _V |-> ( y e. ( A ^m suc k ) |-> ( ( g ` ( y |` k ) ) f ( y ` k ) ) ) ) , { <. (/) , b >. } ) = seq𝜔 ( ( k e. _V , g e. _V |-> ( y e. ( A ^m suc k ) |-> ( ( g ` ( y |` k ) ) f ( y ` k ) ) ) ) , { <. (/) , b >. } )
16		eqid 2622	. . . . . . 7 |- ( x e. U_ n e. om ( A ^m n ) |-> <. dom x , ( (seq𝜔 ( ( k e. _V , g e. _V |-> ( y e. ( A ^m suc k ) |-> ( ( g ` ( y |` k ) ) f ( y ` k ) ) ) ) , { <. (/) , b >. } ) ` dom x ) ` x ) >. ) = ( x e. U_ n e. om ( A ^m n ) |-> <. dom x , ( (seq𝜔 ( ( k e. _V , g e. _V |-> ( y e. ( A ^m suc k ) |-> ( ( g ` ( y |` k ) ) f ( y ` k ) ) ) ) , { <. (/) , b >. } ) ` dom x ) ` x ) >. )
17	11, 14, 5, 15, 16	fseqenlem2 8848	. . . . . 6 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> ( x e. U_ n e. om ( A ^m n ) |-> <. dom x , ( (seq𝜔 ( ( k e. _V , g e. _V |-> ( y e. ( A ^m suc k ) |-> ( ( g ` ( y |` k ) ) f ( y ` k ) ) ) ) , { <. (/) , b >. } ) ` dom x ) ` x ) >. ) : U_ n e. om ( A ^m n ) -1-1-> ( om X. A ) )
18		f1domg 7975	. . . . . 6 |- ( ( om X. A ) e. _V -> ( ( x e. U_ n e. om ( A ^m n ) |-> <. dom x , ( (seq𝜔 ( ( k e. _V , g e. _V |-> ( y e. ( A ^m suc k ) |-> ( ( g ` ( y |` k ) ) f ( y ` k ) ) ) ) , { <. (/) , b >. } ) ` dom x ) ` x ) >. ) : U_ n e. om ( A ^m n ) -1-1-> ( om X. A ) -> U_ n e. om ( A ^m n ) ~<_ ( om X. A ) ) )
19	13, 17, 18	sylc 65	. . . . 5 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> U_ n e. om ( A ^m n ) ~<_ ( om X. A ) )
20		fseqdom 8849	. . . . . 6 |- ( A e. _V -> ( om X. A ) ~<_ U_ n e. om ( A ^m n ) )
21	11, 20	syl 17	. . . . 5 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> ( om X. A ) ~<_ U_ n e. om ( A ^m n ) )
22		sbth 8080	. . . . 5 |- ( ( U_ n e. om ( A ^m n ) ~<_ ( om X. A ) /\ ( om X. A ) ~<_ U_ n e. om ( A ^m n ) ) -> U_ n e. om ( A ^m n ) ~~ ( om X. A ) )
23	19, 21, 22	syl2anc 693	. . . 4 |- ( ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> U_ n e. om ( A ^m n ) ~~ ( om X. A ) )
24	23	exlimivv 1860	. . 3 |- ( E. f E. b ( f : ( A X. A ) -1-1-onto-> A /\ b e. A ) -> U_ n e. om ( A ^m n ) ~~ ( om X. A ) )
25	3, 24	sylbir 225	. 2 |- ( ( E. f f : ( A X. A ) -1-1-onto-> A /\ E. b b e. A ) -> U_ n e. om ( A ^m n ) ~~ ( om X. A ) )
26	1, 2, 25	syl2anb 496	1 |- ( ( ( A X. A ) ~~ A /\ A =/= (/) ) -> U_ n e. om ( A ^m n ) ~~ ( om X. A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   suc csuc 5725   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065  seq_𝜔cseqom 7542   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-map 7859  df-en 7956  df-dom 7957

This theorem is referenced by:  infpwfien  8885