Theorem cc2 7581

Description: Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)

Hypotheses

Ref	Expression
cc2.cc	|- ( ph -> CCHOICE )
cc2.a	|- ( ph -> F Fn om )
cc2.m	|- ( ph -> A. x e. om E. w w e. ( F ` x ) )

Assertion

Ref	Expression
cc2	|- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )

Distinct variable groups:   g, F, n   w, F, x   ph, n

Allowed substitution hints:   ph( x, w, g)

Proof of Theorem cc2

Dummy variables f m v y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cc2.cc	. 2 |- ( ph -> CCHOICE )
2		cc2.a	. 2 |- ( ph -> F Fn om )
3		cc2.m	. . . 4 |- ( ph -> A. x e. om E. w w e. ( F ` x ) )
4		fveq2 5670	. . . . . . 7 |- ( x = y -> ( F ` x ) = ( F ` y ) )
5	4	eleq2d 2302	. . . . . 6 |- ( x = y -> ( w e. ( F ` x ) <-> w e. ( F ` y ) ) )
6	5	exbidv 1874	. . . . 5 |- ( x = y -> ( E. w w e. ( F ` x ) <-> E. w w e. ( F ` y ) ) )
7	6	cbvralv 2778	. . . 4 |- ( A. x e. om E. w w e. ( F ` x ) <-> A. y e. om E. w w e. ( F ` y ) )
8	3, 7	sylib 122	. . 3 |- ( ph -> A. y e. om E. w w e. ( F ` y ) )
9		eleq1w 2293	. . . . 5 |- ( w = v -> ( w e. ( F ` y ) <-> v e. ( F ` y ) ) )
10	9	cbvexv 1968	. . . 4 |- ( E. w w e. ( F ` y ) <-> E. v v e. ( F ` y ) )
11	10	ralbii 2548	. . 3 |- ( A. y e. om E. w w e. ( F ` y ) <-> A. y e. om E. v v e. ( F ` y ) )
12	8, 11	sylib 122	. 2 |- ( ph -> A. y e. om E. v v e. ( F ` y ) )
13		nfcv 2384	. . 3 |- F/_ n ( { m } X. ( F ` m ) )
14		nfcv 2384	. . 3 |- F/_ m ( { n } X. ( F ` n ) )
15		sneq 3700	. . . 4 |- ( m = n -> { m } = { n } )
16		fveq2 5670	. . . 4 |- ( m = n -> ( F ` m ) = ( F ` n ) )
17	15, 16	xpeq12d 4774	. . 3 |- ( m = n -> ( { m } X. ( F ` m ) ) = ( { n } X. ( F ` n ) ) )
18	13, 14, 17	cbvmpt 4205	. 2 |- ( m e. om |-> ( { m } X. ( F ` m ) ) ) = ( n e. om |-> ( { n } X. ( F ` n ) ) )
19		nfcv 2384	. . 3 |- F/_ n ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) )
20		nfcv 2384	. . . 4 |- F/_ m 2nd
21		nfcv 2384	. . . . 5 |- F/_ m f
22		nffvmpt1 5681	. . . . 5 |- F/_ m ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n )
23	21, 22	nffv 5680	. . . 4 |- F/_ m ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) )
24	20, 23	nffv 5680	. . 3 |- F/_ m ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) )
25		2fveq3 5675	. . . 4 |- ( m = n -> ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) = ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) )
26	25	fveq2d 5674	. . 3 |- ( m = n -> ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) ) = ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) ) )
27	19, 24, 26	cbvmpt 4205	. 2 |- ( m e. om |-> ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) ) ) = ( n e. om |-> ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) ) )
28	1, 2, 12, 18, 27	cc2lem 7580	1 |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1541   e. wcel 2203   A.wral 2520   {csn 3689   |-> cmpt 4171   omcom 4712   X. cxp 4747   Fn wfn 5347   ` cfv 5352   2ndc2nd 6333  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:  cc3  7582  acnccim  7586