Theorem cc4f 7635

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 4279	. . . . 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 2624	. . 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 3549	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2556	. . . 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 134	. . 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 7634	. 2 |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) )
12		simprl 535	. . . . . 6 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> f Fn N )
13		elrabi 2979	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } -> ( f ` n ) e. A )
14	13	ralimi 2613	. . . . . . 7 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ( f ` n ) e. A )
15	14	ad2antll 495	. . . . . 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 2392	. . . . . . 7 |- F/_ n N
17		cc4f.a	. . . . . . 7 |- F/_ n A
18		nfcv 2392	. . . . . . 7 |- F/_ n f
19	16, 17, 18	ffnfvf 5867	. . . . . 6 |- ( f : N --> A <-> ( f Fn N /\ A. n e. N ( f ` n ) e. A ) )
20	12, 15, 19	sylanbrc 421	. . . . 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 2982	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } <-> ( ( f ` n ) e. A /\ ch ) )
23	22	simprbi 275	. . . . . . 7 |- ( ( f ` n ) e. { x e. A | ps } -> ch )
24	23	ralimi 2613	. . . . . 6 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ch )
25	24	ad2antll 495	. . . . 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 306	. . . 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 115	. . 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 1933	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   F/_wnfc 2379   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   class class class wbr 4130   omcom 4737   Fn wfn 5372   -->wf 5373   ` cfv 5377   ~~ cen 7020  CCHOICEwacc 7628

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-2nd 6375  df-er 6807  df-en 7023  df-cc 7629

This theorem is used by:  cc4  7636