Theorem cc4n 7627

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7626, 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 2833	. . . . 5 |- ( { x e. A | ps } e. V -> { x e. A | ps } e. _V )
4	3	ralimi 2613	. . . 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 3549	. . . . 5 |- ( E. w w e. { x e. A | ps } <-> E. x e. A ps )
8	7	ralbii 2556	. . . 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 7624	. 2 |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. { x e. A | ps } ) )
12		simprl 535	. . . . 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 2982	. . . . . . . 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 2613	. . . . . 6 |- ( A. n e. N ( f ` n ) e. { x e. A | ps } -> A. n e. N ch )
17	16	ad2antll 495	. . . . 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 1933	. 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 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   class class class wbr 4125   omcom 4732   Fn wfn 5367   ` cfv 5372   ~~ cen 7010  CCHOICEwacc 7618

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-2nd 6365  df-er 6797  df-en 7013  df-cc 7619

This theorem is referenced by:  omctfn  13312