Theorem ccfunen 7096

Description: Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)

Hypotheses

Ref	Expression
ccfunen.cc	|- ( ph -> CCHOICE )
ccfunen.a	|- ( ph -> A ~~ om )
ccfunen.m	|- ( ph -> A. x e. A E. w w e. x )

Assertion

Ref	Expression
ccfunen	|- ( 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)

Proof of Theorem ccfunen

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

Step	Hyp	Ref	Expression
1		ccfunen.a	. . . . . 6 |- ( ph -> A ~~ om )
2		encv 6648	. . . . . 6 |- ( A ~~ om -> ( A e. _V /\ om e. _V ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> ( A e. _V /\ om e. _V ) )
4	3	simpld 111	. . . 4 |- ( ph -> A e. _V )
5		abid2 2261	. . . . . 6 |- { v | v e. u } = u
6		vex 2692	. . . . . 6 |- u e. _V
7	5, 6	eqeltri 2213	. . . . 5 |- { v | v e. u } e. _V
8	7	a1i 9	. . . 4 |- ( ( ph /\ u e. A ) -> { v | v e. u } e. _V )
9	4, 8	opabex3d 6027	. . 3 |- ( ph -> { <. u , v >. | ( u e. A /\ v e. u ) } e. _V )
10		ccfunen.cc	. . . 4 |- ( ph -> CCHOICE )
11		df-cc 7095	. . . 4 |- (CCHOICE <-> A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) )
12	10, 11	sylib 121	. . 3 |- ( ph -> A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) )
13		ccfunen.m	. . . . . 6 |- ( ph -> A. x e. A E. w w e. x )
14		elequ2 1692	. . . . . . . . 9 |- ( x = u -> ( w e. x <-> w e. u ) )
15	14	exbidv 1798	. . . . . . . 8 |- ( x = u -> ( E. w w e. x <-> E. w w e. u ) )
16	15	cbvralv 2657	. . . . . . 7 |- ( A. x e. A E. w w e. x <-> A. u e. A E. w w e. u )
17		elequ1 1691	. . . . . . . . 9 |- ( w = v -> ( w e. u <-> v e. u ) )
18	17	cbvexv 1891	. . . . . . . 8 |- ( E. w w e. u <-> E. v v e. u )
19	18	ralbii 2444	. . . . . . 7 |- ( A. u e. A E. w w e. u <-> A. u e. A E. v v e. u )
20	16, 19	bitri 183	. . . . . 6 |- ( A. x e. A E. w w e. x <-> A. u e. A E. v v e. u )
21	13, 20	sylib 121	. . . . 5 |- ( ph -> A. u e. A E. v v e. u )
22		dmopab3 4760	. . . . 5 |- ( A. u e. A E. v v e. u <-> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
23	21, 22	sylib 121	. . . 4 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
24	23, 1	eqbrtrd 3958	. . 3 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om )
25		dmeq 4747	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> dom y = dom { <. u , v >. | ( u e. A /\ v e. u ) } )
26	25	breq1d 3947	. . . . 5 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( dom y ~~ om <-> dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om ) )
27		sseq2 3126	. . . . . . 7 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( f C_ y <-> f C_ { <. u , v >. | ( u e. A /\ v e. u ) } ) )
28	25	fneq2d 5222	. . . . . . 7 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( f Fn dom y <-> f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) )
29	27, 28	anbi12d 465	. . . . . 6 |- ( 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 ) } ) ) )
30	29	exbidv 1798	. . . . 5 |- ( 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 ) } ) ) )
31	26, 30	imbi12d 233	. . . 4 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) <-> ( dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) ) )
32	31	spcgv 2776	. . 3 |- ( { <. u , v >. | ( u e. A /\ v e. u ) } e. _V -> ( A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) -> ( dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) ) )
33	9, 12, 24, 32	syl3c 63	. 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 ) } ) )
34		simprr 522	. . . . . 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 ) } )
35	23	fneq2d 5222	. . . . . . 7 |- ( ph -> ( f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } <-> f Fn A ) )
36	35	adantr 274	. . . . . 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 ) )
37	34, 36	mpbid 146	. . . . 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 )
38		simplrl 525	. . . . . . . . 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 ) } )
39		fnopfv 5558	. . . . . . . . . 10 |- ( ( f Fn A /\ x e. A ) -> <. x , ( f ` x ) >. e. f )
40	37, 39	sylan 281	. . . . . . . . 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 )
41	38, 40	sseldd 3103	. . . . . . . 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 ) } )
42		vex 2692	. . . . . . . . 9 |- x e. _V
43		vex 2692	. . . . . . . . . 10 |- f e. _V
44	43, 42	fvex 5449	. . . . . . . . 9 |- ( f ` x ) e. _V
45		eleq1 2203	. . . . . . . . . 10 |- ( u = x -> ( u e. A <-> x e. A ) )
46		elequ2 1692	. . . . . . . . . 10 |- ( u = x -> ( v e. u <-> v e. x ) )
47	45, 46	anbi12d 465	. . . . . . . . 9 |- ( u = x -> ( ( u e. A /\ v e. u ) <-> ( x e. A /\ v e. x ) ) )
48		eleq1 2203	. . . . . . . . . 10 |- ( v = ( f ` x ) -> ( v e. x <-> ( f ` x ) e. x ) )
49	48	anbi2d 460	. . . . . . . . 9 |- ( v = ( f ` x ) -> ( ( x e. A /\ v e. x ) <-> ( x e. A /\ ( f ` x ) e. x ) ) )
50	42, 44, 47, 49	opelopab 4201	. . . . . . . 8 |- ( <. x , ( f ` x ) >. e. { <. u , v >. | ( u e. A /\ v e. u ) } <-> ( x e. A /\ ( f ` x ) e. x ) )
51	41, 50	sylib 121	. . . . . . 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 ) )
52	51	simprd 113	. . . . . 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 )
53	52	ralrimiva 2508	. . . . 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 )
54	37, 53	jca 304	. . . 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 ) )
55	54	ex 114	. . 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 ) ) )
56	55	eximdv 1853	. 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 ) ) )
57	33, 56	mpd 13	1 |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   {cab 2126   A.wral 2417   _Vcvv 2689   C_ wss 3076   <.cop 3535   class class class wbr 3937   {copab 3996   omcom 4512   dom cdm 4547   Fn wfn 5126   ` cfv 5131   ~~ cen 6640  CCHOICEwacc 7094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-en 6643  df-cc 7095

This theorem is referenced by:  cc1  7097  cc2lem  7098