Theorem cc4 7211

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 2308	. 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 7210	1 |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982   omcom 4567   -->wf 5184   ` cfv 5188   ~~ cen 6704  CCHOICEwacc 7203

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-2nd 6109  df-er 6501  df-en 6707  df-cc 7204

This theorem is referenced by: (None)