Theorem pwf1oexmid 16899

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 16898	. . . . 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 7550	. . . . . 6 |- 2o ~<_ ~P 1o
5		iunxpconst 4815	. . . . . . . . . . . 12 |- U_ x e. N ( { x } X. 1o ) = ( N X. 1o )
6		df1o2 6674	. . . . . . . . . . . . 13 |- 1o = { (/) }
7	6	xpeq2i 4775	. . . . . . . . . . . 12 |- ( N X. 1o ) = ( N X. { (/) } )
8	1, 5, 7	3eqtri 2259	. . . . . . . . . . 11 |- T = ( N X. { (/) } )
9		peano1 4721	. . . . . . . . . . . 12 |- (/) e. om
10		xpsneng 7086	. . . . . . . . . . . 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 4149	. . . . . . . . . 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 7036	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ T )
15		relen 6992	. . . . . . . . . 10 |- Rel ~~
16		brrelex1 4794	. . . . . . . . . 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 529	. . . . . . . . . 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 5628	. . . . . . . . . 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 7009	. . . . . . . . 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 7037	. . . . . . . 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 7036	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ N )
27		domentr 7044	. . . . . 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 6767	. . . . . . 7 |- 2o e. om
30		nndomo 7131	. . . . . . 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 3259	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N = 2o )
35	26, 34	breqtrd 4140	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ 2o )
36		exmidpw 7181	. . . 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 529	. . . . 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 531	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> N = 2o )
42	40, 41	breqtrd 4140	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ 2o )
43		simprr 533	. . . . . . . . 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 7036	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o ~~ ~P 1o )
46		entr 7037	. . . . . . 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 7140	. . . . . . . 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 7141	. . . . . . . 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 7230	. . . . . 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 600	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 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   (/)c0 3512   ~Pcpw 3674   {csn 3694   U_ciun 3996   class class class wbr 4114  EXMIDwem 4312   omcom 4717   X. cxp 4752   ran crn 4755   Rel wrel 4759   -1-1->wf1 5354   -1-1-onto->wf1o 5356   1oc1o 6653   2oc2o 6654   ~~ cen 6986   ~<_ cdom 6987   Fincfn 6988

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-exmid 4313  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991

This theorem is referenced by: (None)