Theorem cc4 7269

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, 1-May-2024.)

Hypotheses

Ref	Expression
cc4.cc	|- ( ph -> CCHOICE )
cc4.1	|- ( ph -> A e. V )
cc4.2	|- ( ph -> N ~~ om )
cc4.3	|- ( x = ( f ` n ) -> ( ps <-> ch ) )
cc4.m	|- ( ph -> A. n e. N E. x e. A ps )

Assertion

Ref	Expression
cc4	|- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )

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

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

Proof of Theorem cc4

Step	Hyp	Ref	Expression
1		cc4.cc	. 2 |- ( ph -> CCHOICE )
2		cc4.1	. 2 |- ( ph -> A e. V )
3		nfcv 2319	. 2 |- F/_ n A
4		cc4.2	. 2 |- ( ph -> N ~~ om )
5		cc4.3	. 2 |- ( x = ( f ` n ) -> ( ps <-> ch ) )
6		cc4.m	. 2 |- ( ph -> A. n e. N E. x e. A ps )
7	1, 2, 3, 4, 5, 6	cc4f 7268	1 |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4004   omcom 4590   -->wf 5213   ` cfv 5217   ~~ cen 6738  CCHOICEwacc 7261

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-2nd 6142  df-er 6535  df-en 6741  df-cc 7262

This theorem is referenced by: (None)