Theorem ccfunen 7406

Description: Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)

Hypotheses

Ref	Expression
ccfunen.cc	|- ( ph -> CCHOICE )
ccfunen.a	|- ( ph -> A ~~ om )
ccfunen.m	|- ( ph -> A. x e. A E. w w e. x )

Assertion

Ref	Expression
ccfunen	|- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )

Distinct variable groups:   A, f, x   ph, f, x   x, w

Allowed substitution hints:   ph( w)   A( w)

Proof of Theorem ccfunen

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

Step	Hyp	Ref	Expression
1		ccfunen.a	. . . . . 6 |- ( ph -> A ~~ om )
2		encv 6851	. . . . . 6 |- ( A ~~ om -> ( A e. _V /\ om e. _V ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> ( A e. _V /\ om e. _V ) )
4	3	simpld 112	. . . 4 |- ( ph -> A e. _V )
5		abid2 2327	. . . . . 6 |- { v | v e. u } = u
6		vex 2776	. . . . . 6 |- u e. _V
7	5, 6	eqeltri 2279	. . . . 5 |- { v | v e. u } e. _V
8	7	a1i 9	. . . 4 |- ( ( ph /\ u e. A ) -> { v | v e. u } e. _V )
9	4, 8	opabex3d 6224	. . 3 |- ( ph -> { <. u , v >. | ( u e. A /\ v e. u ) } e. _V )
10		ccfunen.cc	. . . 4 |- ( ph -> CCHOICE )
11		df-cc 7405	. . . 4 |- (CCHOICE <-> A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) )
12	10, 11	sylib 122	. . 3 |- ( ph -> A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) )
13		ccfunen.m	. . . . . 6 |- ( ph -> A. x e. A E. w w e. x )
14		elequ2 2182	. . . . . . . . 9 |- ( x = u -> ( w e. x <-> w e. u ) )
15	14	exbidv 1849	. . . . . . . 8 |- ( x = u -> ( E. w w e. x <-> E. w w e. u ) )
16	15	cbvralv 2739	. . . . . . 7 |- ( A. x e. A E. w w e. x <-> A. u e. A E. w w e. u )
17		elequ1 2181	. . . . . . . . 9 |- ( w = v -> ( w e. u <-> v e. u ) )
18	17	cbvexv 1943	. . . . . . . 8 |- ( E. w w e. u <-> E. v v e. u )
19	18	ralbii 2513	. . . . . . 7 |- ( A. u e. A E. w w e. u <-> A. u e. A E. v v e. u )
20	16, 19	bitri 184	. . . . . 6 |- ( A. x e. A E. w w e. x <-> A. u e. A E. v v e. u )
21	13, 20	sylib 122	. . . . 5 |- ( ph -> A. u e. A E. v v e. u )
22		dmopab3 4905	. . . . 5 |- ( A. u e. A E. v v e. u <-> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
23	21, 22	sylib 122	. . . 4 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } = A )
24	23, 1	eqbrtrd 4076	. . 3 |- ( ph -> dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om )
25		dmeq 4892	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> dom y = dom { <. u , v >. | ( u e. A /\ v e. u ) } )
26	25	breq1d 4064	. . . . 5 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( dom y ~~ om <-> dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om ) )
27		sseq2 3221	. . . . . . 7 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( f C_ y <-> f C_ { <. u , v >. | ( u e. A /\ v e. u ) } ) )
28	25	fneq2d 5379	. . . . . . 7 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( f Fn dom y <-> f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) )
29	27, 28	anbi12d 473	. . . . . 6 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( ( f C_ y /\ f Fn dom y ) <-> ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) )
30	29	exbidv 1849	. . . . 5 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( E. f ( f C_ y /\ f Fn dom y ) <-> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) )
31	26, 30	imbi12d 234	. . . 4 |- ( y = { <. u , v >. | ( u e. A /\ v e. u ) } -> ( ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) <-> ( dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) ) )
32	31	spcgv 2864	. . 3 |- ( { <. u , v >. | ( u e. A /\ v e. u ) } e. _V -> ( A. y ( dom y ~~ om -> E. f ( f C_ y /\ f Fn dom y ) ) -> ( dom { <. u , v >. | ( u e. A /\ v e. u ) } ~~ om -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) ) )
33	9, 12, 24, 32	syl3c 63	. 2 |- ( ph -> E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) )
34		simprr 531	. . . . . 6 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } )
35	23	fneq2d 5379	. . . . . . 7 |- ( ph -> ( f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } <-> f Fn A ) )
36	35	adantr 276	. . . . . 6 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> ( f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } <-> f Fn A ) )
37	34, 36	mpbid 147	. . . . 5 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> f Fn A )
38		simplrl 535	. . . . . . . . 9 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> f C_ { <. u , v >. | ( u e. A /\ v e. u ) } )
39		fnopfv 5728	. . . . . . . . . 10 |- ( ( f Fn A /\ x e. A ) -> <. x , ( f ` x ) >. e. f )
40	37, 39	sylan 283	. . . . . . . . 9 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> <. x , ( f ` x ) >. e. f )
41	38, 40	sseldd 3198	. . . . . . . 8 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> <. x , ( f ` x ) >. e. { <. u , v >. | ( u e. A /\ v e. u ) } )
42		vex 2776	. . . . . . . . 9 |- x e. _V
43		vex 2776	. . . . . . . . . 10 |- f e. _V
44	43, 42	fvex 5614	. . . . . . . . 9 |- ( f ` x ) e. _V
45		eleq1 2269	. . . . . . . . . 10 |- ( u = x -> ( u e. A <-> x e. A ) )
46		elequ2 2182	. . . . . . . . . 10 |- ( u = x -> ( v e. u <-> v e. x ) )
47	45, 46	anbi12d 473	. . . . . . . . 9 |- ( u = x -> ( ( u e. A /\ v e. u ) <-> ( x e. A /\ v e. x ) ) )
48		eleq1 2269	. . . . . . . . . 10 |- ( v = ( f ` x ) -> ( v e. x <-> ( f ` x ) e. x ) )
49	48	anbi2d 464	. . . . . . . . 9 |- ( v = ( f ` x ) -> ( ( x e. A /\ v e. x ) <-> ( x e. A /\ ( f ` x ) e. x ) ) )
50	42, 44, 47, 49	opelopab 4331	. . . . . . . 8 |- ( <. x , ( f ` x ) >. e. { <. u , v >. | ( u e. A /\ v e. u ) } <-> ( x e. A /\ ( f ` x ) e. x ) )
51	41, 50	sylib 122	. . . . . . 7 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> ( x e. A /\ ( f ` x ) e. x ) )
52	51	simprd 114	. . . . . 6 |- ( ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) /\ x e. A ) -> ( f ` x ) e. x )
53	52	ralrimiva 2580	. . . . 5 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> A. x e. A ( f ` x ) e. x )
54	37, 53	jca 306	. . . 4 |- ( ( ph /\ ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) ) -> ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
55	54	ex 115	. . 3 |- ( ph -> ( ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) -> ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) )
56	55	eximdv 1904	. 2 |- ( ph -> ( E. f ( f C_ { <. u , v >. | ( u e. A /\ v e. u ) } /\ f Fn dom { <. u , v >. | ( u e. A /\ v e. u ) } ) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) )
57	33, 56	mpd 13	1 |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2177   {cab 2192   A.wral 2485   _Vcvv 2773   C_ wss 3170   <.cop 3641   class class class wbr 4054   {copab 4115   omcom 4651   dom cdm 4688   Fn wfn 5280   ` cfv 5285   ~~ cen 6843  CCHOICEwacc 7404

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 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493

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 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-en 6846  df-cc 7405

This theorem is referenced by:  cc1  7407  cc2lem  7408