Theorem ccfunen 7620

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 7018	. . . . . 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 112	. . . 4 |- ( ph -> A e. _V )
5		abid2 2361	. . . . . 6 |- { v | v e. u } = u
6		vex 2824	. . . . . 6 |- u e. _V
7	5, 6	eqeltri 2311	. . . . 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 6340	. . 3 |- ( ph -> { <. u , v >. | ( u e. A /\ v e. u ) } e. _V )
10		ccfunen.cc	. . . 4 |- ( ph -> CCHOICE )
11		df-cc 7619	. . . 4 |- (CCHOICE <-> A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) )
12	10, 11	sylib 122	. . 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 2214	. . . . . . . . 9 |- ( x = u -> ( w e. x <-> w e. u ) )
15	14	exbidv 1878	. . . . . . . 8 |- ( x = u -> ( E. w w e. x <-> E. w w e. u ) )
16	15	cbvralv 2786	. . . . . . 7 |- ( A. x e. A E. w w e. x <-> A. u e. A E. w w e. u )
17		elequ1 2213	. . . . . . . . 9 |- ( w = v -> ( w e. u <-> v e. u ) )
18	17	cbvexv 1974	. . . . . . . 8 |- ( E. w w e. u <-> E. v v e. u )
19	18	ralbii 2556	. . . . . . 7 |- ( A. u e. A E. w w e. u <-> A. u e. A E. v v e. u )
20	16, 19	bitri 184	. . . . . 6 |- ( A. x e. A E. w w e. x <-> A. u e. A E. v v e. u )
21	13, 20	sylib 122	. . . . 5 |- ( ph -> A. u e. A E. v v e. u )
22		dmopab3 4989	. . . . 5 |- ( A. u e. A E. v v e. u <-> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
23	21, 22	sylib 122	. . . 4 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
24	23, 1	eqbrtrd 4147	. . 3 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om )
25		dmeq 4976	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> dom y = dom { <. u , v >. | ( u e. A /\ v e. u ) } )
26	25	breq1d 4135	. . . . 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 3272	. . . . . . 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 5467	. . . . . . 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 477	. . . . . 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 1878	. . . . 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 234	. . . 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 2912	. . 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 537	. . . . . 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 5467	. . . . . . 7 |- ( ph -> ( f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } <-> f Fn A ) )
36	35	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 ) )
37	34, 36	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 )
38		simplrl 541	. . . . . . . . 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 5829	. . . . . . . . . 10 |- ( ( f Fn A /\ x e. A ) -> <. x , ( f ` x ) >. e. f )
40	37, 39	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 )
41	38, 40	sseldd 3249	. . . . . . . 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 2824	. . . . . . . . 9 |- x e. _V
43		vex 2824	. . . . . . . . . 10 |- f e. _V
44	43, 42	fvex 5710	. . . . . . . . 9 |- ( f ` x ) e. _V
45		eleq1 2301	. . . . . . . . . 10 |- ( u = x -> ( u e. A <-> x e. A ) )
46		elequ2 2214	. . . . . . . . . 10 |- ( u = x -> ( v e. u <-> v e. x ) )
47	45, 46	anbi12d 477	. . . . . . . . 9 |- ( u = x -> ( ( u e. A /\ v e. u ) <-> ( x e. A /\ v e. x ) ) )
48		eleq1 2301	. . . . . . . . . 10 |- ( v = ( f ` x ) -> ( v e. x <-> ( f ` x ) e. x ) )
49	48	anbi2d 468	. . . . . . . . 9 |- ( v = ( f ` x ) -> ( ( x e. A /\ v e. x ) <-> ( x e. A /\ ( f ` x ) e. x ) ) )
50	42, 44, 47, 49	opelopab 4409	. . . . . . . 8 |- ( <. x , ( f ` x ) >. e. { <. u , v >. | ( u e. A /\ v e. u ) } <-> ( x e. A /\ ( f ` x ) e. x ) )
51	41, 50	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 ) )
52	51	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 )
53	52	ralrimiva 2623	. . . . 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 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 ) )
55	54	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 ) ) )
56	55	eximdv 1933	. 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 104   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   A.wral 2528   _Vcvv 2821   C_ wss 3220   <.cop 3708   class class class wbr 4125   {copab 4186   omcom 4732   dom cdm 4769   Fn wfn 5367   ` cfv 5372   ~~ cen 7010  CCHOICEwacc 7618

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-en 7013  df-cc 7619

This theorem is referenced by:  cc1  7621  cc2lem  7622