Theorem pwf1oexmid 16077

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 16076	. . . . 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 7358	. . . . . 6 |- 2o ~<_ ~P 1o
5		iunxpconst 4743	. . . . . . . . . . . 12 |- U_ x e. N ( { x } X. 1o ) = ( N X. 1o )
6		df1o2 6528	. . . . . . . . . . . . 13 |- 1o = { (/) }
7	6	xpeq2i 4704	. . . . . . . . . . . 12 |- ( N X. 1o ) = ( N X. { (/) } )
8	1, 5, 7	3eqtri 2231	. . . . . . . . . . 11 |- T = ( N X. { (/) } )
9		peano1 4650	. . . . . . . . . . . 12 |- (/) e. om
10		xpsneng 6932	. . . . . . . . . . . 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 4086	. . . . . . . . . 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 6888	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ T )
15		relen 6844	. . . . . . . . . 10 |- Rel ~~
16		brrelex1 4722	. . . . . . . . . 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 5543	. . . . . . . . . 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 6861	. . . . . . . . 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 6889	. . . . . . . 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 6888	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ N )
27		domentr 6896	. . . . . 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 6620	. . . . . . 7 |- 2o e. om
30		nndomo 6976	. . . . . . 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 3214	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N = 2o )
35	26, 34	breqtrd 4077	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ 2o )
36		exmidpw 7020	. . . 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 4077	. . . . . . 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 6888	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o ~~ ~P 1o )
46		entr 6889	. . . . . . 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 6984	. . . . . . . 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 6985	. . . . . . . 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 7064	. . . . . 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 596	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 1373   e. wcel 2177   _Vcvv 2773   C_ wss 3170   (/)c0 3464   ~Pcpw 3621   {csn 3638   U_ciun 3933   class class class wbr 4051  EXMIDwem 4246   omcom 4646   X. cxp 4681   ran crn 4684   Rel wrel 4688   -1-1->wf1 5277   -1-1-onto->wf1o 5279   1oc1o 6508   2oc2o 6509   ~~ cen 6838   ~<_ cdom 6839   Fincfn 6840

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-exmid 4247  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-1o 6515  df-2o 6516  df-er 6633  df-en 6841  df-dom 6842  df-fin 6843

This theorem is referenced by: (None)