Theorem pwf1oexmid 13879

Description: An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)

Hypothesis

Ref	Expression
pwle2.t	|- T = U_ x e. N ( { x } X. 1o )

Assertion

Ref	Expression
pwf1oexmid	|- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )

Distinct variable group:   x, N

Allowed substitution hints:   T( x)   G( x)

Proof of Theorem pwf1oexmid

Step	Hyp	Ref	Expression
1		pwle2.t	. . . . . 6 |- T = U_ x e. N ( { x } X. 1o )
2	1	pwle2 13878	. . . . 5 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )
3	2	adantr 274	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N C_ 2o )
4		pw1dom2 7183	. . . . . 6 |- 2o ~<_ ~P 1o
5		iunxpconst 4664	. . . . . . . . . . . 12 |- U_ x e. N ( { x } X. 1o ) = ( N X. 1o )
6		df1o2 6397	. . . . . . . . . . . . 13 |- 1o = { (/) }
7	6	xpeq2i 4625	. . . . . . . . . . . 12 |- ( N X. 1o ) = ( N X. { (/) } )
8	1, 5, 7	3eqtri 2190	. . . . . . . . . . 11 |- T = ( N X. { (/) } )
9		peano1 4571	. . . . . . . . . . . 12 |- (/) e. om
10		xpsneng 6788	. . . . . . . . . . . 12 |- ( ( N e. om /\ (/) e. om ) -> ( N X. { (/) } ) ~~ N )
11	9, 10	mpan2 422	. . . . . . . . . . 11 |- ( N e. om -> ( N X. { (/) } ) ~~ N )
12	8, 11	eqbrtrid 4017	. . . . . . . . . 10 |- ( N e. om -> T ~~ N )
13	12	ad2antrr 480	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T ~~ N )
14	13	ensymd 6749	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ T )
15		relen 6710	. . . . . . . . . 10 |- Rel ~~
16		brrelex1 4643	. . . . . . . . . 10 |- ( ( Rel ~~ /\ T ~~ N ) -> T e. _V )
17	15, 13, 16	sylancr 411	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T e. _V )
18		simplr 520	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> G : T -1-1-> ~P 1o )
19		simpr 109	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ran G = ~P 1o )
20		dff1o5 5441	. . . . . . . . . 10 |- ( G : T -1-1-onto-> ~P 1o <-> ( G : T -1-1-> ~P 1o /\ ran G = ~P 1o ) )
21	18, 19, 20	sylanbrc 414	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> G : T -1-1-onto-> ~P 1o )
22		f1oeng 6723	. . . . . . . . 9 |- ( ( T e. _V /\ G : T -1-1-onto-> ~P 1o ) -> T ~~ ~P 1o )
23	17, 21, 22	syl2anc 409	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T ~~ ~P 1o )
24		entr 6750	. . . . . . . 8 |- ( ( N ~~ T /\ T ~~ ~P 1o ) -> N ~~ ~P 1o )
25	14, 23, 24	syl2anc 409	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ ~P 1o )
26	25	ensymd 6749	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ N )
27		domentr 6757	. . . . . 6 |- ( ( 2o ~<_ ~P 1o /\ ~P 1o ~~ N ) -> 2o ~<_ N )
28	4, 26, 27	sylancr 411	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> 2o ~<_ N )
29		2onn 6489	. . . . . . 7 |- 2o e. om
30		nndomo 6830	. . . . . . 7 |- ( ( 2o e. om /\ N e. om ) -> ( 2o ~<_ N <-> 2o C_ N ) )
31	29, 30	mpan 421	. . . . . 6 |- ( N e. om -> ( 2o ~<_ N <-> 2o C_ N ) )
32	31	ad2antrr 480	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ( 2o ~<_ N <-> 2o C_ N ) )
33	28, 32	mpbid 146	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> 2o C_ N )
34	3, 33	eqssd 3159	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N = 2o )
35	26, 34	breqtrd 4008	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ 2o )
36		exmidpw 6874	. . . 4 |- (EXMID <-> ~P 1o ~~ 2o )
37	35, 36	sylibr 133	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> EXMID )
38	34, 37	jca 304	. 2 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ( N = 2o /\ EXMID ) )
39		simplr 520	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> G : T -1-1-> ~P 1o )
40	12	ad2antrr 480	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ N )
41		simprl 521	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> N = 2o )
42	40, 41	breqtrd 4008	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ 2o )
43		simprr 522	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> EXMID )
44	43, 36	sylib 121	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ~P 1o ~~ 2o )
45	44	ensymd 6749	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o ~~ ~P 1o )
46		entr 6750	. . . . . . 7 |- ( ( T ~~ 2o /\ 2o ~~ ~P 1o ) -> T ~~ ~P 1o )
47	42, 45, 46	syl2anc 409	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ ~P 1o )
48		nnfi 6838	. . . . . . . 8 |- ( 2o e. om -> 2o e. Fin )
49	29, 48	mp1i 10	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o e. Fin )
50		enfi 6839	. . . . . . . 8 |- ( ~P 1o ~~ 2o -> ( ~P 1o e. Fin <-> 2o e. Fin ) )
51	44, 50	syl 14	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ( ~P 1o e. Fin <-> 2o e. Fin ) )
52	49, 51	mpbird 166	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ~P 1o e. Fin )
53		f1finf1o 6912	. . . . . 6 |- ( ( T ~~ ~P 1o /\ ~P 1o e. Fin ) -> ( G : T -1-1-> ~P 1o <-> G : T -1-1-onto-> ~P 1o ) )
54	47, 52, 53	syl2anc 409	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ( G : T -1-1-> ~P 1o <-> G : T -1-1-onto-> ~P 1o ) )
55	39, 54	mpbid 146	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> G : T -1-1-onto-> ~P 1o )
56	55, 20	sylib 121	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ( G : T -1-1-> ~P 1o /\ ran G = ~P 1o ) )
57	56	simprd 113	. 2 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ran G = ~P 1o )
58	38, 57	impbida 586	1 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   (/)c0 3409   ~Pcpw 3559   {csn 3576   U_ciun 3866   class class class wbr 3982  EXMIDwem 4173   omcom 4567   X. cxp 4602   ran crn 4605   Rel wrel 4609   -1-1->wf1 5185   -1-1-onto->wf1o 5187   1oc1o 6377   2oc2o 6378   ~~ cen 6704   ~<_ cdom 6705   Fincfn 6706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-exmid 4174  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709

This theorem is referenced by: (None)