Theorem pwssplit1 19059

Description: Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypotheses

Ref	Expression
pwssplit1.y	|- Y = ( W ^s U )
pwssplit1.z	|- Z = ( W ^s V )
pwssplit1.b	|- B = ( Base ` Y )
pwssplit1.c	|- C = ( Base ` Z )
pwssplit1.f	|- F = ( x e. B |-> ( x |` V ) )

Assertion

Ref	Expression
pwssplit1	|- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> F : B -onto-> C )

Distinct variable groups:   x, Y   x, W   x, U   x, Z   x, V   x, B   x, C   x, X

Allowed substitution hint:   F( x)

Proof of Theorem pwssplit1

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

Step	Hyp	Ref	Expression
1		pwssplit1.y	. . 3 |- Y = ( W ^s U )
2		pwssplit1.z	. . 3 |- Z = ( W ^s V )
3		pwssplit1.b	. . 3 |- B = ( Base ` Y )
4		pwssplit1.c	. . 3 |- C = ( Base ` Z )
5		pwssplit1.f	. . 3 |- F = ( x e. B |-> ( x |` V ) )
6	1, 2, 3, 4, 5	pwssplit0 19058	. 2 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> F : B --> C )
7		simp1 1061	. . . . . . . . 9 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> W e. Mnd )
8		simp2 1062	. . . . . . . . . 10 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> U e. X )
9		simp3 1063	. . . . . . . . . 10 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> V C_ U )
10	8, 9	ssexd 4805	. . . . . . . . 9 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> V e. _V )
11		eqid 2622	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
12	2, 11, 4	pwselbasb 16148	. . . . . . . . 9 |- ( ( W e. Mnd /\ V e. _V ) -> ( a e. C <-> a : V --> ( Base ` W ) ) )
13	7, 10, 12	syl2anc 693	. . . . . . . 8 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> ( a e. C <-> a : V --> ( Base ` W ) ) )
14	13	biimpa 501	. . . . . . 7 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> a : V --> ( Base ` W ) )
15		fvex 6201	. . . . . . . . . 10 |- ( 0g ` W ) e. _V
16	15	fconst 6091	. . . . . . . . 9 |- ( ( U \ V ) X. { ( 0g ` W ) } ) : ( U \ V ) --> { ( 0g ` W ) }
17	16	a1i 11	. . . . . . . 8 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( U \ V ) X. { ( 0g ` W ) } ) : ( U \ V ) --> { ( 0g ` W ) } )
18		simpl1 1064	. . . . . . . . . 10 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> W e. Mnd )
19		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` W ) = ( 0g ` W )
20	11, 19	mndidcl 17308	. . . . . . . . . 10 |- ( W e. Mnd -> ( 0g ` W ) e. ( Base ` W ) )
21	18, 20	syl 17	. . . . . . . . 9 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( 0g ` W ) e. ( Base ` W ) )
22	21	snssd 4340	. . . . . . . 8 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> { ( 0g ` W ) } C_ ( Base ` W ) )
23	17, 22	fssd 6057	. . . . . . 7 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( U \ V ) X. { ( 0g ` W ) } ) : ( U \ V ) --> ( Base ` W ) )
24		disjdif 4040	. . . . . . . 8 |- ( V i^i ( U \ V ) ) = (/)
25	24	a1i 11	. . . . . . 7 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( V i^i ( U \ V ) ) = (/) )
26		fun 6066	. . . . . . 7 |- ( ( ( a : V --> ( Base ` W ) /\ ( ( U \ V ) X. { ( 0g ` W ) } ) : ( U \ V ) --> ( Base ` W ) ) /\ ( V i^i ( U \ V ) ) = (/) ) -> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : ( V u. ( U \ V ) ) --> ( ( Base ` W ) u. ( Base ` W ) ) )
27	14, 23, 25, 26	syl21anc 1325	. . . . . 6 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : ( V u. ( U \ V ) ) --> ( ( Base ` W ) u. ( Base ` W ) ) )
28		simpl3 1066	. . . . . . . 8 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> V C_ U )
29		undif 4049	. . . . . . . 8 |- ( V C_ U <-> ( V u. ( U \ V ) ) = U )
30	28, 29	sylib 208	. . . . . . 7 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( V u. ( U \ V ) ) = U )
31		unidm 3756	. . . . . . . 8 |- ( ( Base ` W ) u. ( Base ` W ) ) = ( Base ` W )
32	31	a1i 11	. . . . . . 7 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( Base ` W ) u. ( Base ` W ) ) = ( Base ` W ) )
33	30, 32	feq23d 6040	. . . . . 6 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : ( V u. ( U \ V ) ) --> ( ( Base ` W ) u. ( Base ` W ) ) <-> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : U --> ( Base ` W ) ) )
34	27, 33	mpbid 222	. . . . 5 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : U --> ( Base ` W ) )
35		simpl2 1065	. . . . . 6 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> U e. X )
36	1, 11, 3	pwselbasb 16148	. . . . . 6 |- ( ( W e. Mnd /\ U e. X ) -> ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) e. B <-> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : U --> ( Base ` W ) ) )
37	18, 35, 36	syl2anc 693	. . . . 5 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) e. B <-> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) : U --> ( Base ` W ) ) )
38	34, 37	mpbird 247	. . . 4 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) e. B )
39	5	fvtresfn 6284	. . . . . 6 |- ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) e. B -> ( F ` ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) ) = ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) |` V ) )
40	38, 39	syl 17	. . . . 5 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( F ` ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) ) = ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) |` V ) )
41		resundir 5411	. . . . . . 7 |- ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) |` V ) = ( ( a |` V ) u. ( ( ( U \ V ) X. { ( 0g ` W ) } ) |` V ) )
42		ffn 6045	. . . . . . . . 9 |- ( a : V --> ( Base ` W ) -> a Fn V )
43		fnresdm 6000	. . . . . . . . 9 |- ( a Fn V -> ( a |` V ) = a )
44	14, 42, 43	3syl 18	. . . . . . . 8 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( a |` V ) = a )
45		incom 3805	. . . . . . . . . 10 |- ( ( U \ V ) i^i V ) = ( V i^i ( U \ V ) )
46	45, 24	eqtri 2644	. . . . . . . . 9 |- ( ( U \ V ) i^i V ) = (/)
47		fnconstg 6093	. . . . . . . . . . 11 |- ( ( 0g ` W ) e. _V -> ( ( U \ V ) X. { ( 0g ` W ) } ) Fn ( U \ V ) )
48	15, 47	ax-mp 5	. . . . . . . . . 10 |- ( ( U \ V ) X. { ( 0g ` W ) } ) Fn ( U \ V )
49		fnresdisj 6001	. . . . . . . . . 10 |- ( ( ( U \ V ) X. { ( 0g ` W ) } ) Fn ( U \ V ) -> ( ( ( U \ V ) i^i V ) = (/) <-> ( ( ( U \ V ) X. { ( 0g ` W ) } ) |` V ) = (/) ) )
50	48, 49	mp1i 13	. . . . . . . . 9 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( ( U \ V ) i^i V ) = (/) <-> ( ( ( U \ V ) X. { ( 0g ` W ) } ) |` V ) = (/) ) )
51	46, 50	mpbii 223	. . . . . . . 8 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( ( U \ V ) X. { ( 0g ` W ) } ) |` V ) = (/) )
52	44, 51	uneq12d 3768	. . . . . . 7 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( a |` V ) u. ( ( ( U \ V ) X. { ( 0g ` W ) } ) |` V ) ) = ( a u. (/) ) )
53	41, 52	syl5eq 2668	. . . . . 6 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) |` V ) = ( a u. (/) ) )
54		un0 3967	. . . . . 6 |- ( a u. (/) ) = a
55	53, 54	syl6eq 2672	. . . . 5 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) |` V ) = a )
56	40, 55	eqtr2d 2657	. . . 4 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> a = ( F ` ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) ) )
57		fveq2 6191	. . . . . 6 |- ( b = ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) -> ( F ` b ) = ( F ` ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) ) )
58	57	eqeq2d 2632	. . . . 5 |- ( b = ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) -> ( a = ( F ` b ) <-> a = ( F ` ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) ) ) )
59	58	rspcev 3309	. . . 4 |- ( ( ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) e. B /\ a = ( F ` ( a u. ( ( U \ V ) X. { ( 0g ` W ) } ) ) ) ) -> E. b e. B a = ( F ` b ) )
60	38, 56, 59	syl2anc 693	. . 3 |- ( ( ( W e. Mnd /\ U e. X /\ V C_ U ) /\ a e. C ) -> E. b e. B a = ( F ` b ) )
61	60	ralrimiva 2966	. 2 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> A. a e. C E. b e. B a = ( F ` b ) )
62		dffo3 6374	. 2 |- ( F : B -onto-> C <-> ( F : B --> C /\ A. a e. C E. b e. B a = ( F ` b ) ) )
63	6, 61, 62	sylanbrc 698	1 |- ( ( W e. Mnd /\ U e. X /\ V C_ U ) -> F : B -onto-> C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   |` cres 5116   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   ^_s cpws 16107   Mndcmnd 17294

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-riota 6611  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  pwslnmlem2  37663