Theorem acfun 7206

Description: A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)

Hypotheses

Ref	Expression
acfun.ac	|- ( ph -> CHOICE )
acfun.a	|- ( ph -> A e. V )
acfun.m	|- ( ph -> A. x e. A E. w w e. x )

Assertion

Ref	Expression
acfun	|- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )

Distinct variable groups:   A, f, x   ph, f, x   x, w

Allowed substitution hints:   ph( w)   A( w)   V( x, w, f)

Proof of Theorem acfun

Dummy variables u v y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		acfun.a	. . . . 5 |- ( ph -> A e. V )
2	1	elexd 2751	. . . 4 |- ( ph -> A e. _V )
3		abid2 2298	. . . . . 6 |- { v | v e. u } = u
4		vex 2741	. . . . . 6 |- u e. _V
5	3, 4	eqeltri 2250	. . . . 5 |- { v | v e. u } e. _V
6	5	a1i 9	. . . 4 |- ( ( ph /\ u e. A ) -> { v | v e. u } e. _V )
7	2, 6	opabex3d 6122	. . 3 |- ( ph -> { <. u , v >. | ( u e. A /\ v e. u ) } e. _V )
8		acfun.ac	. . . 4 |- ( ph -> CHOICE )
9		df-ac 7205	. . . 4 |- (CHOICE <-> A. y E. f ( f C_ y /\ f Fn dom y ) )
10	8, 9	sylib 122	. . 3 |- ( ph -> A. y E. f ( f C_ y /\ f Fn dom y ) )
11		sseq2 3180	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( f C_ y <-> f C_ { <. u , v >. | ( u e. A /\ v e. u ) } ) )
12		dmeq 4828	. . . . . . 7 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> dom y = dom { <. u , v >. | ( u e. A /\ v e. u ) } )
13	12	fneq2d 5308	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( f Fn dom y <-> f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) )
14	11, 13	anbi12d 473	. . . . 5 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( ( f C_ y /\ f Fn dom y ) <-> ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) )
15	14	exbidv 1825	. . . 4 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( E. f ( f C_ y /\ f Fn dom y ) <-> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) )
16	15	spcgv 2825	. . 3 |- ( { <. u , v >. | ( u e. A /\ v e. u ) } e. _V -> ( A. y E. f ( f C_ y /\ f Fn dom y ) -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) )
17	7, 10, 16	sylc 62	. 2 |- ( ph -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) )
18		simprr 531	. . . . . 6 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } )
19		acfun.m	. . . . . . . . . 10 |- ( ph -> A. x e. A E. w w e. x )
20		elequ2 2153	. . . . . . . . . . . . 13 |- ( x = u -> ( w e. x <-> w e. u ) )
21	20	exbidv 1825	. . . . . . . . . . . 12 |- ( x = u -> ( E. w w e. x <-> E. w w e. u ) )
22	21	cbvralv 2704	. . . . . . . . . . 11 |- ( A. x e. A E. w w e. x <-> A. u e. A E. w w e. u )
23		elequ1 2152	. . . . . . . . . . . . 13 |- ( w = v -> ( w e. u <-> v e. u ) )
24	23	cbvexv 1918	. . . . . . . . . . . 12 |- ( E. w w e. u <-> E. v v e. u )
25	24	ralbii 2483	. . . . . . . . . . 11 |- ( A. u e. A E. w w e. u <-> A. u e. A E. v v e. u )
26	22, 25	bitri 184	. . . . . . . . . 10 |- ( A. x e. A E. w w e. x <-> A. u e. A E. v v e. u )
27	19, 26	sylib 122	. . . . . . . . 9 |- ( ph -> A. u e. A E. v v e. u )
28		dmopab3 4841	. . . . . . . . 9 |- ( A. u e. A E. v v e. u <-> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
29	27, 28	sylib 122	. . . . . . . 8 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
30	29	fneq2d 5308	. . . . . . 7 |- ( ph -> ( f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } <-> f Fn A ) )
31	30	adantr 276	. . . . . 6 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> ( f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } <-> f Fn A ) )
32	18, 31	mpbid 147	. . . . 5 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> f Fn A )
33		simplrl 535	. . . . . . . . 9 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> f C_ { <. u , v >. | ( u e. A /\ v e. u ) } )
34		fnopfv 5647	. . . . . . . . . 10 |- ( ( f Fn A /\ x e. A ) -> <. x , ( f ` x ) >. e. f )
35	32, 34	sylan 283	. . . . . . . . 9 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> <. x , ( f ` x ) >. e. f )
36	33, 35	sseldd 3157	. . . . . . . 8 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> <. x , ( f ` x ) >. e. { <. u , v >. | ( u e. A /\ v e. u ) } )
37		vex 2741	. . . . . . . . 9 |- x e. _V
38		vex 2741	. . . . . . . . . 10 |- f e. _V
39	38, 37	fvex 5536	. . . . . . . . 9 |- ( f ` x ) e. _V
40		eleq1 2240	. . . . . . . . . 10 |- ( u = x -> ( u e. A <-> x e. A ) )
41		elequ2 2153	. . . . . . . . . 10 |- ( u = x -> ( v e. u <-> v e. x ) )
42	40, 41	anbi12d 473	. . . . . . . . 9 |- ( u = x -> ( ( u e. A /\ v e. u ) <-> ( x e. A /\ v e. x ) ) )
43		eleq1 2240	. . . . . . . . . 10 |- ( v = ( f ` x ) -> ( v e. x <-> ( f ` x ) e. x ) )
44	43	anbi2d 464	. . . . . . . . 9 |- ( v = ( f ` x ) -> ( ( x e. A /\ v e. x ) <-> ( x e. A /\ ( f ` x ) e. x ) ) )
45	37, 39, 42, 44	opelopab 4272	. . . . . . . 8 |- ( <. x , ( f ` x ) >. e. { <. u , v >. | ( u e. A /\ v e. u ) } <-> ( x e. A /\ ( f ` x ) e. x ) )
46	36, 45	sylib 122	. . . . . . 7 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> ( x e. A /\ ( f ` x ) e. x ) )
47	46	simprd 114	. . . . . 6 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> ( f ` x ) e. x )
48	47	ralrimiva 2550	. . . . 5 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> A. x e. A ( f ` x ) e. x )
49	32, 48	jca 306	. . . 4 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
50	49	ex 115	. . 3 |- ( ph -> ( ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) -> ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) )
51	50	eximdv 1880	. 2 |- ( ph -> ( E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) )
52	17, 51	mpd 13	1 |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   A.wral 2455   _Vcvv 2738   C_ wss 3130   <.cop 3596   {copab 4064   dom cdm 4627   Fn wfn 5212   ` cfv 5217  CHOICEwac 7204

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-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

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  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-ac 7205

This theorem is referenced by:  exmidaclem  7207