Theorem ccfunen 7526

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 6958	. . . . . 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 2353	. . . . . 6 |- { v | v e. u } = u
6		vex 2806	. . . . . 6 |- u e. _V
7	5, 6	eqeltri 2304	. . . . 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 6292	. . 3 |- ( ph -> { <. u , v >. | ( u e. A /\ v e. u ) } e. _V )
10		ccfunen.cc	. . . 4 |- ( ph -> CCHOICE )
11		df-cc 7525	. . . 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 2207	. . . . . . . . 9 |- ( x = u -> ( w e. x <-> w e. u ) )
15	14	exbidv 1873	. . . . . . . 8 |- ( x = u -> ( E. w w e. x <-> E. w w e. u ) )
16	15	cbvralv 2768	. . . . . . 7 |- ( A. x e. A E. w w e. x <-> A. u e. A E. w w e. u )
17		elequ1 2206	. . . . . . . . 9 |- ( w = v -> ( w e. u <-> v e. u ) )
18	17	cbvexv 1967	. . . . . . . 8 |- ( E. w w e. u <-> E. v v e. u )
19	18	ralbii 2539	. . . . . . 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 4950	. . . . 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 4115	. . 3 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om )
25		dmeq 4937	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> dom y = dom { <. u , v >. | ( u e. A /\ v e. u ) } )
26	25	breq1d 4103	. . . . 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 3252	. . . . . . 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 5428	. . . . . . 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 473	. . . . . 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 1873	. . . . 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 2894	. . 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 533	. . . . . 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 5428	. . . . . . 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 537	. . . . . . . . 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 5785	. . . . . . . . . 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 3229	. . . . . . . 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 2806	. . . . . . . . 9 |- x e. _V
43		vex 2806	. . . . . . . . . 10 |- f e. _V
44	43, 42	fvex 5668	. . . . . . . . 9 |- ( f ` x ) e. _V
45		eleq1 2294	. . . . . . . . . 10 |- ( u = x -> ( u e. A <-> x e. A ) )
46		elequ2 2207	. . . . . . . . . 10 |- ( u = x -> ( v e. u <-> v e. x ) )
47	45, 46	anbi12d 473	. . . . . . . . 9 |- ( u = x -> ( ( u e. A /\ v e. u ) <-> ( x e. A /\ v e. x ) ) )
48		eleq1 2294	. . . . . . . . . 10 |- ( v = ( f ` x ) -> ( v e. x <-> ( f ` x ) e. x ) )
49	48	anbi2d 464	. . . . . . . . 9 |- ( v = ( f ` x ) -> ( ( x e. A /\ v e. x ) <-> ( x e. A /\ ( f ` x ) e. x ) ) )
50	42, 44, 47, 49	opelopab 4372	. . . . . . . 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 2606	. . . . 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 1928	. 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 1396   = wceq 1398   E.wex 1541   e. wcel 2202   {cab 2217   A.wral 2511   _Vcvv 2803   C_ wss 3201   <.cop 3676   class class class wbr 4093   {copab 4154   omcom 4694   dom cdm 4731   Fn wfn 5328   ` cfv 5333   ~~ cen 6950  CCHOICEwacc 7524

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 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

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  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-en 6953  df-cc 7525

This theorem is referenced by:  cc1  7527  cc2lem  7528