Theorem nninfomnilem 16727

Description: Lemma for nninfomni 16728. (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 7363	. . 3 |- ℕ∞ e. _V
2		isomnimap 7379	. . 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 6882	. . . . . 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 16722	. . . . . . 7 |- E : ( 2o ^m ℕ∞ ) -->ℕ∞
7	6	ffvelcdmi 5789	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E ` r ) e. ℕ∞ )
8	4, 7	ffvelcdmd 5791	. . . . 5 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. 2o )
9		df2o3 6640	. . . . 5 |- 2o = { (/) , 1o }
10	8, 9	eleqtrdi 2324	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. { (/) , 1o } )
11		elpri 3696	. . . 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 5657	. . . . . . 7 |- ( p = ( E ` r ) -> ( ( r ` p ) = (/) <-> ( r ` ( E ` r ) ) = (/) ) )
14	13	rspcev 2911	. . . . . 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 16726	. . . . 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 794	. . 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 2591	1 |- ℕ∞ e. Omni

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   _Vcvv 2803   (/)c0 3496   ifcif 3607   {cpr 3674   |-> cmpt 4155   suc csuc 4468   omcom 4694   ` cfv 5333  (class class class)co 6028   1oc1o 6618   2oc2o 6619   ^m cmap 6860  ℕ_∞xnninf 7361  Omnicomni 7376

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 619  ax-in2 620  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-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  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-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  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-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-map 6862  df-nninf 7362  df-omni 7377

This theorem is referenced by:  nninfomni  16728