Theorem cc2 7529

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 5648	. . . . . . 7 |- ( x = y -> ( F ` x ) = ( F ` y ) )
5	4	eleq2d 2301	. . . . . 6 |- ( x = y -> ( w e. ( F ` x ) <-> w e. ( F ` y ) ) )
6	5	exbidv 1873	. . . . 5 |- ( x = y -> ( E. w w e. ( F ` x ) <-> E. w w e. ( F ` y ) ) )
7	6	cbvralv 2768	. . . 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 2292	. . . . 5 |- ( w = v -> ( w e. ( F ` y ) <-> v e. ( F ` y ) ) )
10	9	cbvexv 1967	. . . 4 |- ( E. w w e. ( F ` y ) <-> E. v v e. ( F ` y ) )
11	10	ralbii 2539	. . 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 2375	. . 3 |- F/_ n ( { m } X. ( F ` m ) )
14		nfcv 2375	. . 3 |- F/_ m ( { n } X. ( F ` n ) )
15		sneq 3684	. . . 4 |- ( m = n -> { m } = { n } )
16		fveq2 5648	. . . 4 |- ( m = n -> ( F ` m ) = ( F ` n ) )
17	15, 16	xpeq12d 4756	. . 3 |- ( m = n -> ( { m } X. ( F ` m ) ) = ( { n } X. ( F ` n ) ) )
18	13, 14, 17	cbvmpt 4189	. 2 |- ( m e. om |-> ( { m } X. ( F ` m ) ) ) = ( n e. om |-> ( { n } X. ( F ` n ) ) )
19		nfcv 2375	. . 3 |- F/_ n ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) )
20		nfcv 2375	. . . 4 |- F/_ m 2nd
21		nfcv 2375	. . . . 5 |- F/_ m f
22		nffvmpt1 5659	. . . . 5 |- F/_ m ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n )
23	21, 22	nffv 5658	. . . 4 |- F/_ m ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) )
24	20, 23	nffv 5658	. . 3 |- F/_ m ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) )
25		2fveq3 5653	. . . 4 |- ( m = n -> ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` m ) ) = ( f ` ( ( m e. om |-> ( { m } X. ( F ` m ) ) ) ` n ) ) )
26	25	fveq2d 5652	. . 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 4189	. 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 7528	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 2202   A.wral 2511   {csn 3673   |-> cmpt 4155   omcom 4694   X. cxp 4729   Fn wfn 5328   ` cfv 5333   2ndc2nd 6311  CCHOICEwacc 7524

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-2nd 6313  df-er 6745  df-en 6953  df-cc 7525

This theorem is referenced by:  cc3  7530  acnccim  7534