Theorem imasbas 12728

Description: The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)

Hypotheses

Ref	Expression
imasbas.u	|- ( ph -> U = ( F "s R ) )
imasbas.v	|- ( ph -> V = ( Base ` R ) )
imasbas.f	|- ( ph -> F : V -onto-> B )
imasbas.r	|- ( ph -> R e. Z )

Assertion

Ref	Expression
imasbas	|- ( ph -> B = ( Base ` U ) )

Proof of Theorem imasbas

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasbas.u	. . . 4 |- ( ph -> U = ( F "s R ) )
2		imasbas.v	. . . 4 |- ( ph -> V = ( Base ` R ) )
3		eqid 2177	. . . 4 |- ( +g ` R ) = ( +g ` R )
4		eqid 2177	. . . 4 |- ( .r ` R ) = ( .r ` R )
5		eqid 2177	. . . 4 |- ( .s ` R ) = ( .s ` R )
6		eqidd 2178	. . . 4 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
7		eqidd 2178	. . . 4 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
8		imasbas.f	. . . 4 |- ( ph -> F : V -onto-> B )
9		imasbas.r	. . . 4 |- ( ph -> R e. Z )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	imasival 12727	. . 3 |- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
11	10	fveq1d 5518	. 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } ` ( Base ` ndx ) ) )
12		basendxnn 12518	. . . . . 6 |- ( Base ` ndx ) e. NN
13		basfn 12520	. . . . . . . . 9 |- Base Fn _V
14	9	elexd 2751	. . . . . . . . 9 |- ( ph -> R e. _V )
15		funfvex 5533	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
16	15	funfni 5317	. . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
17	13, 14, 16	sylancr 414	. . . . . . . 8 |- ( ph -> ( Base ` R ) e. _V )
18	2, 17	eqeltrd 2254	. . . . . . 7 |- ( ph -> V e. _V )
19		focdmex 6116	. . . . . . 7 |- ( V e. _V -> ( F : V -onto-> B -> B e. _V ) )
20	18, 8, 19	sylc 62	. . . . . 6 |- ( ph -> B e. _V )
21		opexg 4229	. . . . . 6 |- ( ( ( Base ` ndx ) e. NN /\ B e. _V ) -> <. ( Base ` ndx ) , B >. e. _V )
22	12, 20, 21	sylancr 414	. . . . 5 |- ( ph -> <. ( Base ` ndx ) , B >. e. _V )
23		plusgndxnn 12570	. . . . . 6 |- ( +g ` ndx ) e. NN
24		fof 5439	. . . . . . . . . . . . . . . 16 |- ( F : V -onto-> B -> F : V --> B )
25	8, 24	syl 14	. . . . . . . . . . . . . . 15 |- ( ph -> F : V --> B )
26	25, 18	fexd 5747	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
27		vex 2741	. . . . . . . . . . . . . 14 |- p e. _V
28		fvexg 5535	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
29	26, 27, 28	sylancl 413	. . . . . . . . . . . . 13 |- ( ph -> ( F ` p ) e. _V )
30		vex 2741	. . . . . . . . . . . . . 14 |- q e. _V
31		fvexg 5535	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
32	26, 30, 31	sylancl 413	. . . . . . . . . . . . 13 |- ( ph -> ( F ` q ) e. _V )
33		opexg 4229	. . . . . . . . . . . . 13 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
34	29, 32, 33	syl2anc 411	. . . . . . . . . . . 12 |- ( ph -> <. ( F ` p ) , ( F ` q ) >. e. _V )
35	27	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> p e. _V )
36		plusgslid 12571	. . . . . . . . . . . . . . . 16 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
37	36	slotex 12489	. . . . . . . . . . . . . . 15 |- ( R e. Z -> ( +g ` R ) e. _V )
38	9, 37	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> ( +g ` R ) e. _V )
39	30	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> q e. _V )
40		ovexg 5909	. . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
41	35, 38, 39, 40	syl3anc 1238	. . . . . . . . . . . . 13 |- ( ph -> ( p ( +g ` R ) q ) e. _V )
42		fvexg 5535	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ ( p ( +g ` R ) q ) e. _V ) -> ( F ` ( p ( +g ` R ) q ) ) e. _V )
43	26, 41, 42	syl2anc 411	. . . . . . . . . . . 12 |- ( ph -> ( F ` ( p ( +g ` R ) q ) ) e. _V )
44		opexg 4229	. . . . . . . . . . . 12 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p ( +g ` R ) q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
45	34, 43, 44	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
46		snexg 4185	. . . . . . . . . . 11 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
47	45, 46	syl 14	. . . . . . . . . 10 |- ( ph -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
48	47	ralrimivw 2551	. . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
49		iunexg 6120	. . . . . . . . 9 |- ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
50	18, 48, 49	syl2anc 411	. . . . . . . 8 |- ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
51	50	ralrimivw 2551	. . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
52		iunexg 6120	. . . . . . 7 |- ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
53	18, 51, 52	syl2anc 411	. . . . . 6 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
54		opexg 4229	. . . . . 6 |- ( ( ( +g ` ndx ) e. NN /\ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V )
55	23, 53, 54	sylancr 414	. . . . 5 |- ( ph -> <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V )
56		mulrslid 12590	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
57	56	simpri 113	. . . . . 6 |- ( .r ` ndx ) e. NN
58	56	slotex 12489	. . . . . . . . . . . . . . 15 |- ( R e. Z -> ( .r ` R ) e. _V )
59	9, 58	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> ( .r ` R ) e. _V )
60		ovexg 5909	. . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
61	35, 59, 39, 60	syl3anc 1238	. . . . . . . . . . . . 13 |- ( ph -> ( p ( .r ` R ) q ) e. _V )
62		fvexg 5535	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ ( p ( .r ` R ) q ) e. _V ) -> ( F ` ( p ( .r ` R ) q ) ) e. _V )
63	26, 61, 62	syl2anc 411	. . . . . . . . . . . 12 |- ( ph -> ( F ` ( p ( .r ` R ) q ) ) e. _V )
64		opexg 4229	. . . . . . . . . . . 12 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p ( .r ` R ) q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
65	34, 63, 64	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
66		snexg 4185	. . . . . . . . . . 11 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
67	65, 66	syl 14	. . . . . . . . . 10 |- ( ph -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
68	67	ralrimivw 2551	. . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
69		iunexg 6120	. . . . . . . . 9 |- ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
70	18, 68, 69	syl2anc 411	. . . . . . . 8 |- ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
71	70	ralrimivw 2551	. . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
72		iunexg 6120	. . . . . . 7 |- ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
73	18, 71, 72	syl2anc 411	. . . . . 6 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
74		opexg 4229	. . . . . 6 |- ( ( ( .r ` ndx ) e. NN /\ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V )
75	57, 73, 74	sylancr 414	. . . . 5 |- ( ph -> <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V )
76		tpexg 4445	. . . . 5 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V /\ <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
77	22, 55, 75, 76	syl3anc 1238	. . . 4 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
78	10, 77	eqeltrd 2254	. . 3 |- ( ph -> U e. _V )
79		baseid 12516	. . 3 |- Base = Slot ( Base ` ndx )
80	78, 79, 12	strndxid 12490	. 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( Base ` U ) )
81	12	a1i 9	. . 3 |- ( ph -> ( Base ` ndx ) e. NN )
82		basendxnplusgndx 12583	. . . 4 |- ( Base ` ndx ) =/= ( +g ` ndx )
83	82	a1i 9	. . 3 |- ( ph -> ( Base ` ndx ) =/= ( +g ` ndx ) )
84		basendxnmulrndx 12592	. . . 4 |- ( Base ` ndx ) =/= ( .r ` ndx )
85	84	a1i 9	. . 3 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
86		fvtp1g 5725	. . 3 |- ( ( ( ( Base ` ndx ) e. NN /\ B e. _V ) /\ ( ( Base ` ndx ) =/= ( +g ` ndx ) /\ ( Base ` ndx ) =/= ( .r ` ndx ) ) ) -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } ` ( Base ` ndx ) ) = B )
87	81, 20, 83, 85, 86	syl22anc 1239	. 2 |- ( ph -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } ` ( Base ` ndx ) ) = B )
88	11, 80, 87	3eqtr3rd 2219	1 |- ( ph -> B = ( Base ` U ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   _Vcvv 2738   {csn 3593   {ctp 3595   <.cop 3596   U_ciun 3887   Fn wfn 5212   -->wf 5213   -onto->wfo 5215   ` cfv 5217  (class class class)co 5875   NNcn 8919   ndxcnx 12459  Slot cslot 12461   Basecbs 12462   +g cplusg 12536   .rcmulr 12537   .scvsca 12540   "_s cimas 12720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-tp 3601  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-mulr 12550  df-iimas 12723

This theorem is referenced by:  qusbas  12747