Theorem imasbas 13470

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 2231	. . . 4 |- ( +g ` R ) = ( +g ` R )
4		eqid 2231	. . . 4 |- ( .r ` R ) = ( .r ` R )
5		eqid 2231	. . . 4 |- ( .s ` R ) = ( .s ` R )
6		eqidd 2232	. . . 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 2232	. . . 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 13469	. . 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 5650	. 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 13218	. . . . . 6 |- ( Base ` ndx ) e. NN
13		basfn 13221	. . . . . . . . 9 |- Base Fn _V
14	9	elexd 2817	. . . . . . . . 9 |- ( ph -> R e. _V )
15		funfvex 5665	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
16	15	funfni 5439	. . . . . . . . 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 2308	. . . . . . 7 |- ( ph -> V e. _V )
19		focdmex 6286	. . . . . . 7 |- ( V e. _V -> ( F : V -onto-> B -> B e. _V ) )
20	18, 8, 19	sylc 62	. . . . . 6 |- ( ph -> B e. _V )
21		opexg 4326	. . . . . 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 13274	. . . . . 6 |- ( +g ` ndx ) e. NN
24		fof 5568	. . . . . . . . . . . . . . . 16 |- ( F : V -onto-> B -> F : V --> B )
25	8, 24	syl 14	. . . . . . . . . . . . . . 15 |- ( ph -> F : V --> B )
26	25, 18	fexd 5894	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
27		vex 2806	. . . . . . . . . . . . . 14 |- p e. _V
28		fvexg 5667	. . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . . 14 |- q e. _V
31		fvexg 5667	. . . . . . . . . . . . . 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 4326	. . . . . . . . . . . . 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 13275	. . . . . . . . . . . . . . . 16 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
37	36	slotex 13189	. . . . . . . . . . . . . . 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 6062	. . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
41	35, 38, 39, 40	syl3anc 1274	. . . . . . . . . . . . 13 |- ( ph -> ( p ( +g ` R ) q ) e. _V )
42		fvexg 5667	. . . . . . . . . . . . 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 4326	. . . . . . . . . . . 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 4280	. . . . . . . . . . 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 2607	. . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
49		iunexg 6290	. . . . . . . . 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 2607	. . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
52		iunexg 6290	. . . . . . 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 4326	. . . . . 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 13295	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
57	56	simpri 113	. . . . . 6 |- ( .r ` ndx ) e. NN
58	56	slotex 13189	. . . . . . . . . . . . . . 15 |- ( R e. Z -> ( .r ` R ) e. _V )
59	9, 58	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> ( .r ` R ) e. _V )
60		ovexg 6062	. . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
61	35, 59, 39, 60	syl3anc 1274	. . . . . . . . . . . . 13 |- ( ph -> ( p ( .r ` R ) q ) e. _V )
62		fvexg 5667	. . . . . . . . . . . . 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 4326	. . . . . . . . . . . 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 4280	. . . . . . . . . . 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 2607	. . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
69		iunexg 6290	. . . . . . . . 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 2607	. . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
72		iunexg 6290	. . . . . . 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 4326	. . . . . 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 4547	. . . . 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 1274	. . . 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 2308	. . 3 |- ( ph -> U e. _V )
79		baseid 13216	. . 3 |- Base = Slot ( Base ` ndx )
80	78, 79, 12	strndxid 13190	. 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( Base ` U ) )
81	12	a1i 9	. . 3 |- ( ph -> ( Base ` ndx ) e. NN )
82		basendxnplusgndx 13288	. . . 4 |- ( Base ` ndx ) =/= ( +g ` ndx )
83	82	a1i 9	. . 3 |- ( ph -> ( Base ` ndx ) =/= ( +g ` ndx ) )
84		basendxnmulrndx 13297	. . . 4 |- ( Base ` ndx ) =/= ( .r ` ndx )
85	84	a1i 9	. . 3 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
86		fvtp1g 5870	. . 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 1275	. 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 2273	1 |- ( ph -> B = ( Base ` U ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   _Vcvv 2803   {csn 3673   {ctp 3675   <.cop 3676   U_ciun 3975   Fn wfn 5328   -->wf 5329   -onto->wfo 5331   ` cfv 5333  (class class class)co 6028   NNcn 9202   ndxcnx 13159  Slot cslot 13161   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   .scvsca 13244   "_s cimas 13462

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-mulr 13254  df-iimas 13465

This theorem is referenced by:  qusbas  13490  imasmnd2  13615  imasgrp2  13777  imasabl  14003  imasrng  14050  imasring  14158