Theorem cc4n 7403

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7402, 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 2785	. . . . 5 |- ( { x e. A | ps } e. V -> { x e. A | ps } e. _V )
4	3	ralimi 2570	. . . 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 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		cc4n.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	. . . . 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 2933	. . . . . . . 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 2570	. . . . . 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 1904	. 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 1373   E.wex 1516   e. wcel 2177   A.wral 2485   E.wrex 2486   {crab 2489   _Vcvv 2773   class class class wbr 4051   omcom 4646   Fn wfn 5275   ` 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:  omctfn  12889