Theorem cc4n 7382

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7381, 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 2782	. . . . 5 |- ( { x e. A | ps } e. V -> { x e. A | ps } e. _V )
4	3	ralimi 2568	. . . 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 3487	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2511	. . . 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 7379	. 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 2928	. . . . . . . 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 2568	. . . . . 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 1902	. 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 1372   E.wex 1514   e. wcel 2175   A.wral 2483   E.wrex 2484   {crab 2487   _Vcvv 2771   class class class wbr 4043   omcom 4637   Fn wfn 5265   ` cfv 5270   ~~ cen 6824  CCHOICEwacc 7373

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-2nd 6226  df-er 6619  df-en 6827  df-cc 7374

This theorem is referenced by:  omctfn  12756