Theorem nninfdcinf 7288

Description: The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ_∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)

Hypotheses

Ref	Expression
nninfdcinf.w	|- ( ph -> om e. WOmni )
nninfdcinf.n	|- ( ph -> N e. ℕ∞ )

Assertion

Ref	Expression
nninfdcinf	|- ( ph -> DECID N = ( i e. om |-> 1o ) )

Distinct variable groups:   i, N   ph, i

Proof of Theorem nninfdcinf

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5588	. . . . . 6 |- ( f = N -> ( f ` x ) = ( N ` x ) )
2	1	eqeq1d 2215	. . . . 5 |- ( f = N -> ( ( f ` x ) = 1o <-> ( N ` x ) = 1o ) )
3	2	ralbidv 2507	. . . 4 |- ( f = N -> ( A. x e. om ( f ` x ) = 1o <-> A. x e. om ( N ` x ) = 1o ) )
4	3	dcbid 840	. . 3 |- ( f = N -> (DECID A. x e. om ( f ` x ) = 1o <-> DECID A. x e. om ( N ` x ) = 1o ) )
5		nninfdcinf.w	. . . 4 |- ( ph -> om e. WOmni )
6	5	elexd 2787	. . . . 5 |- ( ph -> om e. _V )
7		iswomnimap 7283	. . . . 5 |- ( om e. _V -> ( om e. WOmni <-> A. f e. ( 2o ^m om )DECID A. x e. om ( f ` x ) = 1o ) )
8	6, 7	syl 14	. . . 4 |- ( ph -> ( om e. WOmni <-> A. f e. ( 2o ^m om )DECID A. x e. om ( f ` x ) = 1o ) )
9	5, 8	mpbid 147	. . 3 |- ( ph -> A. f e. ( 2o ^m om )DECID A. x e. om ( f ` x ) = 1o )
10		nninfdcinf.n	. . . . 5 |- ( ph -> N e. ℕ∞ )
11		nninff 7239	. . . . 5 |- ( N e. ℕ∞ -> N : om --> 2o )
12	10, 11	syl 14	. . . 4 |- ( ph -> N : om --> 2o )
13		2onn 6620	. . . . . 6 |- 2o e. om
14	13	elexi 2786	. . . . 5 |- 2o e. _V
15		omex 4649	. . . . 5 |- om e. _V
16	14, 15	elmap 6777	. . . 4 |- ( N e. ( 2o ^m om ) <-> N : om --> 2o )
17	12, 16	sylibr 134	. . 3 |- ( ph -> N e. ( 2o ^m om ) )
18	4, 9, 17	rspcdva 2886	. 2 |- ( ph -> DECID A. x e. om ( N ` x ) = 1o )
19	12	ffnd 5436	. . . 4 |- ( ph -> N Fn om )
20		eqidd 2207	. . . 4 |- ( x = i -> 1o = 1o )
21		1onn 6619	. . . . 5 |- 1o e. om
22	21	a1i 9	. . . 4 |- ( ( ph /\ x e. om ) -> 1o e. om )
23	21	a1i 9	. . . 4 |- ( ( ph /\ i e. om ) -> 1o e. om )
24	19, 20, 22, 23	fnmptfvd 5697	. . 3 |- ( ph -> ( N = ( i e. om |-> 1o ) <-> A. x e. om ( N ` x ) = 1o ) )
25	24	dcbid 840	. 2 |- ( ph -> (DECID N = ( i e. om |-> 1o ) <-> DECID A. x e. om ( N ` x ) = 1o ) )
26	18, 25	mpbird 167	1 |- ( ph -> DECID N = ( i e. om |-> 1o ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 836   = wceq 1373   e. wcel 2177   A.wral 2485   _Vcvv 2773   |-> cmpt 4113   omcom 4646   -->wf 5276   ` cfv 5280  (class class class)co 5957   1oc1o 6508   2oc2o 6509   ^m cmap 6748  ℕ_∞xnninf 7236  WOmnicwomni 7280

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-in1 615  ax-in2 616  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-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  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-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1o 6515  df-2o 6516  df-map 6750  df-nninf 7237  df-womni 7281

This theorem is referenced by:  nninfinfwlpo  7297