Theorem cc4 7549

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 2375	. 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 7548	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 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093   omcom 4694   -->wf 5329   ` cfv 5333   ~~ cen 6950  CCHOICEwacc 7541

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-2nd 6313  df-er 6745  df-en 6953  df-cc 7542

This theorem is referenced by: (None)