Theorem nninfomnilem 15508

Description: Lemma for nninfomni 15509. (Contributed by Jim Kingdon, 10-Aug-2022.)

Hypothesis

Ref	Expression
nninfsel.e	|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )

Assertion

Ref	Expression
nninfomnilem	|- ℕ∞ e. Omni

Distinct variable groups:   i, E, k, n   i, q, k, n

Allowed substitution hint:   E( q)

Proof of Theorem nninfomnilem

Dummy variables p r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfex 7180	. . 3 |- ℕ∞ e. _V
2		isomnimap 7196	. . 3 |- (ℕ∞ e. _V -> (ℕ∞ e. Omni <-> A. r e. ( 2o ^m ℕ∞ ) ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) ) )
3	1, 2	ax-mp 5	. 2 |- (ℕ∞ e. Omni <-> A. r e. ( 2o ^m ℕ∞ ) ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) )
4		elmapi 6724	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> r :ℕ∞ --> 2o )
5		nninfsel.e	. . . . . . . 8 |- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )
6	5	nninfself 15503	. . . . . . 7 |- E : ( 2o ^m ℕ∞ ) -->ℕ∞
7	6	ffvelcdmi 5692	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E ` r ) e. ℕ∞ )
8	4, 7	ffvelcdmd 5694	. . . . 5 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. 2o )
9		df2o3 6483	. . . . 5 |- 2o = { (/) , 1o }
10	8, 9	eleqtrdi 2286	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. { (/) , 1o } )
11		elpri 3641	. . . 4 |- ( ( r ` ( E ` r ) ) e. { (/) , 1o } -> ( ( r ` ( E ` r ) ) = (/) \/ ( r ` ( E ` r ) ) = 1o ) )
12	10, 11	syl 14	. . 3 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = (/) \/ ( r ` ( E ` r ) ) = 1o ) )
13		fveqeq2 5563	. . . . . . 7 |- ( p = ( E ` r ) -> ( ( r ` p ) = (/) <-> ( r ` ( E ` r ) ) = (/) ) )
14	13	rspcev 2864	. . . . . 6 |- ( ( ( E ` r ) e. ℕ∞ /\ ( r ` ( E ` r ) ) = (/) ) -> E. p e. ℕ∞ ( r ` p ) = (/) )
15	14	ex 115	. . . . 5 |- ( ( E ` r ) e. ℕ∞ -> ( ( r ` ( E ` r ) ) = (/) -> E. p e. ℕ∞ ( r ` p ) = (/) ) )
16	7, 15	syl 14	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = (/) -> E. p e. ℕ∞ ( r ` p ) = (/) ) )
17		simpl 109	. . . . . 6 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> r e. ( 2o ^m ℕ∞ ) )
18		simpr 110	. . . . . 6 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> ( r ` ( E ` r ) ) = 1o )
19	5, 17, 18	nninfsel 15507	. . . . 5 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> A. p e. ℕ∞ ( r ` p ) = 1o )
20	19	ex 115	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = 1o -> A. p e. ℕ∞ ( r ` p ) = 1o ) )
21	16, 20	orim12d 787	. . 3 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( ( r ` ( E ` r ) ) = (/) \/ ( r ` ( E ` r ) ) = 1o ) -> ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) ) )
22	12, 21	mpd 13	. 2 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) )
23	3, 22	mprgbir 2552	1 |- ℕ∞ e. Omni

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   (/)c0 3446   ifcif 3557   {cpr 3619   |-> cmpt 4090   suc csuc 4396   omcom 4622   ` cfv 5254  (class class class)co 5918   1oc1o 6462   2oc2o 6463   ^m cmap 6702  ℕ_∞xnninf 7178  Omnicomni 7193

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1o 6469  df-2o 6470  df-map 6704  df-nninf 7179  df-omni 7194

This theorem is referenced by:  nninfomni  15509