Theorem cc4 7600

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 2386	. 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 7599	1 |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4114   omcom 4717   -->wf 5353   ` cfv 5357   ~~ cen 6986  CCHOICEwacc 7592

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-2nd 6348  df-er 6780  df-en 6989  df-cc 7593

This theorem is referenced by: (None)