Theorem cc4f 7380

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 4186	. . . . 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 2579	. . 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 3487	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2511	. . . 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 7379	. 2 |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) )
12		simprl 529	. . . . . 6 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> f Fn N )
13		elrabi 2925	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } -> ( f ` n ) e. A )
14	13	ralimi 2568	. . . . . . 7 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ( f ` n ) e. A )
15	14	ad2antll 491	. . . . . 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 2347	. . . . . . 7 |- F/_ n N
17		cc4f.a	. . . . . . 7 |- F/_ n A
18		nfcv 2347	. . . . . . 7 |- F/_ n f
19	16, 17, 18	ffnfvf 5738	. . . . . 6 |- ( f : N --> A <-> ( f Fn N /\ A. n e. N ( f ` n ) e. A ) )
20	12, 15, 19	sylanbrc 417	. . . . 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 2928	. . . . . . . 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 2568	. . . . . 6 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ch )
25	24	ad2antll 491	. . . . 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 1902	. 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 104   <-> wb 105   = wceq 1372   E.wex 1514   e. wcel 2175   F/_wnfc 2334   A.wral 2483   E.wrex 2484   {crab 2487   _Vcvv 2771   class class class wbr 4043   omcom 4637   Fn wfn 5265   -->wf 5266   ` cfv 5270   ~~ cen 6824  CCHOICEwacc 7373

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-2nd 6226  df-er 6619  df-en 6827  df-cc 7374

This theorem is referenced by:  cc4  7381