Theorem pwf1oexmid 16536

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 16535	. . . . 5 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )
3	2	adantr 276	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N C_ 2o )
4		pw1dom2 7435	. . . . . 6 |- 2o ~<_ ~P 1o
5		iunxpconst 4784	. . . . . . . . . . . 12 |- U_ x e. N ( { x } X. 1o ) = ( N X. 1o )
6		df1o2 6591	. . . . . . . . . . . . 13 |- 1o = { (/) }
7	6	xpeq2i 4744	. . . . . . . . . . . 12 |- ( N X. 1o ) = ( N X. { (/) } )
8	1, 5, 7	3eqtri 2254	. . . . . . . . . . 11 |- T = ( N X. { (/) } )
9		peano1 4690	. . . . . . . . . . . 12 |- (/) e. om
10		xpsneng 7001	. . . . . . . . . . . 12 |- ( ( N e. om /\ (/) e. om ) -> ( N X. { (/) } ) ~~ N )
11	9, 10	mpan2 425	. . . . . . . . . . 11 |- ( N e. om -> ( N X. { (/) } ) ~~ N )
12	8, 11	eqbrtrid 4121	. . . . . . . . . 10 |- ( N e. om -> T ~~ N )
13	12	ad2antrr 488	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T ~~ N )
14	13	ensymd 6952	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ T )
15		relen 6908	. . . . . . . . . 10 |- Rel ~~
16		brrelex1 4763	. . . . . . . . . 10 |- ( ( Rel ~~ /\ T ~~ N ) -> T e. _V )
17	15, 13, 16	sylancr 414	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T e. _V )
18		simplr 528	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> G : T -1-1-> ~P 1o )
19		simpr 110	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ran G = ~P 1o )
20		dff1o5 5589	. . . . . . . . . 10 |- ( G : T -1-1-onto-> ~P 1o <-> ( G : T -1-1-> ~P 1o /\ ran G = ~P 1o ) )
21	18, 19, 20	sylanbrc 417	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> G : T -1-1-onto-> ~P 1o )
22		f1oeng 6925	. . . . . . . . 9 |- ( ( T e. _V /\ G : T -1-1-onto-> ~P 1o ) -> T ~~ ~P 1o )
23	17, 21, 22	syl2anc 411	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T ~~ ~P 1o )
24		entr 6953	. . . . . . . 8 |- ( ( N ~~ T /\ T ~~ ~P 1o ) -> N ~~ ~P 1o )
25	14, 23, 24	syl2anc 411	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ ~P 1o )
26	25	ensymd 6952	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ N )
27		domentr 6960	. . . . . 6 |- ( ( 2o ~<_ ~P 1o /\ ~P 1o ~~ N ) -> 2o ~<_ N )
28	4, 26, 27	sylancr 414	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> 2o ~<_ N )
29		2onn 6684	. . . . . . 7 |- 2o e. om
30		nndomo 7045	. . . . . . 7 |- ( ( 2o e. om /\ N e. om ) -> ( 2o ~<_ N <-> 2o C_ N ) )
31	29, 30	mpan 424	. . . . . 6 |- ( N e. om -> ( 2o ~<_ N <-> 2o C_ N ) )
32	31	ad2antrr 488	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ( 2o ~<_ N <-> 2o C_ N ) )
33	28, 32	mpbid 147	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> 2o C_ N )
34	3, 33	eqssd 3242	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N = 2o )
35	26, 34	breqtrd 4112	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ 2o )
36		exmidpw 7093	. . . 4 |- (EXMID <-> ~P 1o ~~ 2o )
37	35, 36	sylibr 134	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> EXMID )
38	34, 37	jca 306	. 2 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ( N = 2o /\ EXMID ) )
39		simplr 528	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> G : T -1-1-> ~P 1o )
40	12	ad2antrr 488	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ N )
41		simprl 529	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> N = 2o )
42	40, 41	breqtrd 4112	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ 2o )
43		simprr 531	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> EXMID )
44	43, 36	sylib 122	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ~P 1o ~~ 2o )
45	44	ensymd 6952	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o ~~ ~P 1o )
46		entr 6953	. . . . . . 7 |- ( ( T ~~ 2o /\ 2o ~~ ~P 1o ) -> T ~~ ~P 1o )
47	42, 45, 46	syl2anc 411	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ ~P 1o )
48		nnfi 7054	. . . . . . . 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 7055	. . . . . . . 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 167	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ~P 1o e. Fin )
53		f1finf1o 7137	. . . . . 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 411	. . . . 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 147	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> G : T -1-1-onto-> ~P 1o )
56	55, 20	sylib 122	. . 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 114	. 2 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ran G = ~P 1o )
58	38, 57	impbida 598	1 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   C_ wss 3198   (/)c0 3492   ~Pcpw 3650   {csn 3667   U_ciun 3968   class class class wbr 4086  EXMIDwem 4282   omcom 4686   X. cxp 4721   ran crn 4724   Rel wrel 4728   -1-1->wf1 5321   -1-1-onto->wf1o 5323   1oc1o 6570   2oc2o 6571   ~~ cen 6902   ~<_ cdom 6903   Fincfn 6904

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-exmid 4283  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907

This theorem is referenced by: (None)