Theorem nninfomnilem 16343

Description: Lemma for nninfomni 16344. (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 7284	. . 3 |- ℕ∞ e. _V
2		isomnimap 7300	. . 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 6815	. . . . . 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 16338	. . . . . . 7 |- E : ( 2o ^m ℕ∞ ) -->ℕ∞
7	6	ffvelcdmi 5768	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E ` r ) e. ℕ∞ )
8	4, 7	ffvelcdmd 5770	. . . . 5 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. 2o )
9		df2o3 6574	. . . . 5 |- 2o = { (/) , 1o }
10	8, 9	eleqtrdi 2322	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. { (/) , 1o } )
11		elpri 3689	. . . 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 5635	. . . . . . 7 |- ( p = ( E ` r ) -> ( ( r ` p ) = (/) <-> ( r ` ( E ` r ) ) = (/) ) )
14	13	rspcev 2907	. . . . . 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 16342	. . . . 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 791	. . 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 2588	1 |- ℕ∞ e. Omni

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2799   (/)c0 3491   ifcif 3602   {cpr 3667   |-> cmpt 4144   suc csuc 4455   omcom 4681   ` cfv 5317  (class class class)co 6000   1oc1o 6553   2oc2o 6554   ^m cmap 6793  ℕ_∞xnninf 7282  Omnicomni 7297

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1o 6560  df-2o 6561  df-map 6795  df-nninf 7283  df-omni 7298

This theorem is referenced by:  nninfomni  16344