Theorem cc4f 7401

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 4195	. . . . 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 2581	. . 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 3492	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2513	. . . 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 7400	. 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 2930	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } -> ( f ` n ) e. A )
14	13	ralimi 2570	. . . . . . 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 2349	. . . . . . 7 |- F/_ n N
17		cc4f.a	. . . . . . 7 |- F/_ n A
18		nfcv 2349	. . . . . . 7 |- F/_ n f
19	16, 17, 18	ffnfvf 5752	. . . . . 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 2933	. . . . . . . 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 2570	. . . . . 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 1904	. 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 1373   E.wex 1516   e. wcel 2177   F/_wnfc 2336   A.wral 2485   E.wrex 2486   {crab 2489   _Vcvv 2773   class class class wbr 4051   omcom 4646   Fn wfn 5275   -->wf 5276   ` cfv 5280   ~~ cen 6838  CCHOICEwacc 7394

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-2nd 6240  df-er 6633  df-en 6841  df-cc 7395

This theorem is referenced by:  cc4  7402