Theorem cc4f 7101

Description: Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)

Hypotheses

Ref	Expression
cc4f.cc	|- ( ph -> CCHOICE )
cc4f.1	|- ( ph -> A e. V )
cc4f.a	|- F/_ n A
cc4f.2	|- ( ph -> N ~~ om )
cc4f.3	|- ( x = ( f ` n ) -> ( ps <-> ch ) )
cc4f.m	|- ( ph -> A. n e. N E. x e. A ps )

Assertion

Ref	Expression
cc4f	|- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )

Distinct variable groups:   A, f, x   f, N, n   ch, x   ph, f, n   ps, f   x, n

Allowed substitution hints:   ph( x)   ps( x, n)   ch( f, n)   A( n)   N( x)   V( x, f, n)

Proof of Theorem cc4f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cc4f.cc	. . 3 |- ( ph -> CCHOICE )
2		cc4f.1	. . . . 5 |- ( ph -> A e. V )
3		rabexg 4079	. . . . 5 |- ( A e. V -> { x e. A | ps } e. _V )
4	2, 3	syl 14	. . . 4 |- ( ph -> { x e. A | ps } e. _V )
5	4	ralrimivw 2509	. . 3 |- ( ph -> A. n e. N { x e. A | ps } e. _V )
6		cc4f.m	. . . 4 |- ( ph -> A. n e. N E. x e. A ps )
7		rabn0m 3395	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2444	. . . 4 |- ( A. n e. N E. w w e. { x e. A | ps } <-> A. n e. N E. x e. A ps )
9	6, 8	sylibr 133	. . 3 |- ( ph -> A. n e. N E. w w e. { x e. A | ps } )
10		cc4f.2	. . 3 |- ( ph -> N ~~ om )
11	1, 5, 9, 10	cc3 7100	. 2 |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) )
12		simprl 521	. . . . . 6 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> f Fn N )
13		elrabi 2841	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } -> ( f ` n ) e. A )
14	13	ralimi 2498	. . . . . . 7 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ( f ` n ) e. A )
15	14	ad2antll 483	. . . . . 6 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> A. n e. N ( f ` n ) e. A )
16		nfcv 2282	. . . . . . 7 |- F/_ n N
17		cc4f.a	. . . . . . 7 |- F/_ n A
18		nfcv 2282	. . . . . . 7 |- F/_ n f
19	16, 17, 18	ffnfvf 5587	. . . . . 6 |- ( f : N --> A <-> ( f Fn N /\ A. n e. N ( f ` n ) e. A ) )
20	12, 15, 19	sylanbrc 414	. . . . 5 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> f : N --> A )
21		cc4f.3	. . . . . . . . 9 |- ( x = ( f ` n ) -> ( ps <-> ch ) )
22	21	elrab 2844	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } <-> ( ( f ` n ) e. A /\ ch ) )
23	22	simprbi 273	. . . . . . 7 |- ( ( f ` n ) e. { x e. A | ps } -> ch )
24	23	ralimi 2498	. . . . . 6 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ch )
25	24	ad2antll 483	. . . . 5 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> A. n e. N ch )
26	20, 25	jca 304	. . . 4 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> ( f : N --> A /\ A. n e. N ch ) )
27	26	ex 114	. . 3 |- ( ph -> ( ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) -> ( f : N --> A /\ A. n e. N ch ) ) )
28	27	eximdv 1853	. 2 |- ( ph -> ( E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) -> E. f ( f : N --> A /\ A. n e. N ch ) ) )
29	11, 28	mpd 13	1 |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   F/_wnfc 2269   A.wral 2417   E.wrex 2418   {crab 2421   _Vcvv 2689   class class class wbr 3937   omcom 4512   Fn wfn 5126   -->wf 5127   ` 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  ax-iinf 4510

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-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-iom 4513  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-2nd 6047  df-er 6437  df-en 6643  df-cc 7095

This theorem is referenced by:  cc4  7102