Theorem cc4n 7585

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7584, the hypotheses only require an A(n) for each value of n, not a single set A which suffices for every n e. om. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)

Hypotheses

Ref	Expression
cc4n.cc	|- ( ph -> CCHOICE )
cc4n.1	|- ( ph -> A. n e. N { x e. A | ps } e. V )
cc4n.2	|- ( ph -> N ~~ om )
cc4n.3	|- ( x = ( f ` n ) -> ( ps <-> ch ) )
cc4n.m	|- ( ph -> A. n e. N E. x e. A ps )

Assertion

Ref	Expression
cc4n	|- ( ph -> E. f ( f Fn N /\ 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 cc4n

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cc4n.cc	. . 3 |- ( ph -> CCHOICE )
2		cc4n.1	. . . 4 |- ( ph -> A. n e. N { x e. A | ps } e. V )
3		elex 2825	. . . . 5 |- ( { x e. A | ps } e. V -> { x e. A | ps } e. _V )
4	3	ralimi 2605	. . . 4 |- ( A. n e. N { x e. A | ps } e. V -> A. n e. N { x e. A | ps } e. _V )
5	2, 4	syl 14	. . 3 |- ( ph -> A. n e. N { x e. A | ps } e. _V )
6		cc4n.m	. . . 4 |- ( ph -> A. n e. N E. x e. A ps )
7		rabn0m 3536	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2548	. . . 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		cc4n.2	. . 3 |- ( ph -> N ~~ om )
11	1, 5, 9, 10	cc3 7582	. 2 |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) )
12		simprl 531	. . . . 5 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> f Fn N )
13		cc4n.3	. . . . . . . . 9 |- ( x = ( f ` n ) -> ( ps <-> ch ) )
14	13	elrab 2973	. . . . . . . 8 |- ( ( f ` n ) e. { x e. A | ps } <-> ( ( f ` n ) e. A /\ ch ) )
15	14	simprbi 275	. . . . . . 7 |- ( ( f ` n ) e. { x e. A | ps } -> ch )
16	15	ralimi 2605	. . . . . 6 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ch )
17	16	ad2antll 491	. . . . 5 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> A. n e. N ch )
18	12, 17	jca 306	. . . 4 |- ( ( ph /\ ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) ) -> ( f Fn N /\ A. n e. N ch ) )
19	18	ex 115	. . 3 |- ( ph -> ( ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) -> ( f Fn N /\ A. n e. N ch ) ) )
20	19	eximdv 1929	. 2 |- ( ph -> ( E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) -> E. f ( f Fn N /\ A. n e. N ch ) ) )
21	11, 20	mpd 13	1 |- ( ph -> E. f ( f Fn N /\ A. n e. N ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   _Vcvv 2813   class class class wbr 4109   omcom 4712   Fn wfn 5347   ` cfv 5352   ~~ cen 6973  CCHOICEwacc 7576

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-2nd 6335  df-er 6767  df-en 6976  df-cc 7577

This theorem is referenced by:  omctfn  13194