Theorem cc2 7623

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 5690	. . . . . . 7 |- ( x = y -> ( F ` x ) = ( F ` y ) )
5	4	eleq2d 2308	. . . . . 6 |- ( x = y -> ( w e. ( F ` x ) <-> w e. ( F ` y ) ) )
6	5	exbidv 1878	. . . . 5 |- ( x = y -> ( E. w w e. ( F ` x ) <-> E. w w e. ( F ` y ) ) )
7	6	cbvralv 2786	. . . 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 2299	. . . . 5 |- ( w = v -> ( w e. ( F ` y ) <-> v e. ( F ` y ) ) )
10	9	cbvexv 1974	. . . 4 |- ( E. w w e. ( F ` y ) <-> E. v v e. ( F ` y ) )
11	10	ralbii 2556	. . 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 2392	. . 3 |- F/_ n ( { m } X. ( F ` m ) )
14		nfcv 2392	. . 3 |- F/_ m ( { n } X. ( F ` n ) )
15		sneq 3716	. . . 4 |- ( m = n -> { m } = { n } )
16		fveq2 5690	. . . 4 |- ( m = n -> ( F ` m ) = ( F ` n ) )
17	15, 16	xpeq12d 4794	. . 3 |- ( m = n -> ( { m } X. ( F ` m ) ) = ( { n } X. ( F ` n ) ) )
18	13, 14, 17	cbvmpt 4221	. 2 |- ( m e. om |-> ( { m } X. ( F ` m ) ) ) = ( n e. om |-> ( { n } X. ( F ` n ) ) )
19		nfcv 2392	. . 3 |- F/_ n ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) )
20		nfcv 2392	. . . 4 |- F/_ m 2nd
21		nfcv 2392	. . . . 5 |- F/_ m f
22		nffvmpt1 5701	. . . . 5 |- F/_ m ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n )
23	21, 22	nffv 5700	. . . 4 |- F/_ m ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) )
24	20, 23	nffv 5700	. . 3 |- F/_ m ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) )
25		2fveq3 5695	. . . 4 |- ( m = n -> ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) = ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) )
26	25	fveq2d 5694	. . 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 4221	. 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 7622	1 |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   A.wral 2528   {csn 3705   |-> cmpt 4187   omcom 4732   X. cxp 4767   Fn wfn 5367   ` cfv 5372   2ndc2nd 6363  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:  cc3  7624  acnccim  7628