Theorem cc4n 7305

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7304, 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 2763	. . . . 5 |- ( { x e. A | ps } e. V -> { x e. A | ps } e. _V )
4	3	ralimi 2553	. . . 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 3465	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2496	. . . 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 7302	. 2 |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) )
12		simprl 529	. . . . 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 2908	. . . . . . . 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 2553	. . . . . 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 1891	. 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 1364   E.wex 1503   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   _Vcvv 2752   class class class wbr 4021   omcom 4610   Fn wfn 5233   ` cfv 5238   ~~ cen 6768  CCHOICEwacc 7296

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-iinf 4608

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-2nd 6170  df-er 6563  df-en 6771  df-cc 7297

This theorem is referenced by:  omctfn  12505