Theorem pwf1oexmid 16760

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 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o)

Assertion

Ref	Expression
pwf1oexmid	⊢ ((𝑁 ∈ ω ∧ 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 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o)
2	1	pwle2 16759	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o)
3	2	adantr 276	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑁 ⊆ 2o)
4		pw1dom2 7536	. . . . . 6 ⊢ 2o ≼ 𝒫 1o
5		iunxpconst 4809	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) = (𝑁 × 1o)
6		df1o2 6660	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
7	6	xpeq2i 4769	. . . . . . . . . . . 12 ⊢ (𝑁 × 1o) = (𝑁 × {∅})
8	1, 5, 7	3eqtri 2257	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑁 × {∅})
9		peano1 4715	. . . . . . . . . . . 12 ⊢ ∅ ∈ ω
10		xpsneng 7072	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ ∅ ∈ ω) → (𝑁 × {∅}) ≈ 𝑁)
11	9, 10	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑁 × {∅}) ≈ 𝑁)
12	8, 11	eqbrtrid 4143	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → 𝑇 ≈ 𝑁)
13	12	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑇 ≈ 𝑁)
14	13	ensymd 7022	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑁 ≈ 𝑇)
15		relen 6978	. . . . . . . . . 10 ⊢ Rel ≈
16		brrelex1 4788	. . . . . . . . . 10 ⊢ ((Rel ≈ ∧ 𝑇 ≈ 𝑁) → 𝑇 ∈ V)
17	15, 13, 16	sylancr 414	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑇 ∈ V)
18		simplr 529	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝐺:𝑇–1-1→𝒫 1o)
19		simpr 110	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → ran 𝐺 = 𝒫 1o)
20		dff1o5 5622	. . . . . . . . . 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 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝐺:𝑇–1-1-onto→𝒫 1o)
22		f1oeng 6995	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝐺:𝑇–1-1-onto→𝒫 1o) → 𝑇 ≈ 𝒫 1o)
23	17, 21, 22	syl2anc 411	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑇 ≈ 𝒫 1o)
24		entr 7023	. . . . . . . 8 ⊢ ((𝑁 ≈ 𝑇 ∧ 𝑇 ≈ 𝒫 1o) → 𝑁 ≈ 𝒫 1o)
25	14, 23, 24	syl2anc 411	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑁 ≈ 𝒫 1o)
26	25	ensymd 7022	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝒫 1o ≈ 𝑁)
27		domentr 7030	. . . . . 6 ⊢ ((2o ≼ 𝒫 1o ∧ 𝒫 1o ≈ 𝑁) → 2o ≼ 𝑁)
28	4, 26, 27	sylancr 414	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 2o ≼ 𝑁)
29		2onn 6753	. . . . . . 7 ⊢ 2o ∈ ω
30		nndomo 7117	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 𝑁 ∈ ω) → (2o ≼ 𝑁 ↔ 2o ⊆ 𝑁))
31	29, 30	mpan 424	. . . . . 6 ⊢ (𝑁 ∈ ω → (2o ≼ 𝑁 ↔ 2o ⊆ 𝑁))
32	31	ad2antrr 488	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → (2o ≼ 𝑁 ↔ 2o ⊆ 𝑁))
33	28, 32	mpbid 147	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 2o ⊆ 𝑁)
34	3, 33	eqssd 3254	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑁 = 2o)
35	26, 34	breqtrd 4134	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝒫 1o ≈ 2o)
36		exmidpw 7167	. . . 4 ⊢ (EXMID ↔ 𝒫 1o ≈ 2o)
37	35, 36	sylibr 134	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → EXMID)
38	34, 37	jca 306	. 2 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → (𝑁 = 2o ∧ EXMID))
39		simplr 529	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝐺:𝑇–1-1→𝒫 1o)
40	12	ad2antrr 488	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝑇 ≈ 𝑁)
41		simprl 531	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝑁 = 2o)
42	40, 41	breqtrd 4134	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝑇 ≈ 2o)
43		simprr 533	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → EXMID)
44	43, 36	sylib 122	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝒫 1o ≈ 2o)
45	44	ensymd 7022	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 2o ≈ 𝒫 1o)
46		entr 7023	. . . . . . 7 ⊢ ((𝑇 ≈ 2o ∧ 2o ≈ 𝒫 1o) → 𝑇 ≈ 𝒫 1o)
47	42, 45, 46	syl2anc 411	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝑇 ≈ 𝒫 1o)
48		nnfi 7126	. . . . . . . 8 ⊢ (2o ∈ ω → 2o ∈ Fin)
49	29, 48	mp1i 10	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 2o ∈ Fin)
50		enfi 7127	. . . . . . . 8 ⊢ (𝒫 1o ≈ 2o → (𝒫 1o ∈ Fin ↔ 2o ∈ Fin))
51	44, 50	syl 14	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → (𝒫 1o ∈ Fin ↔ 2o ∈ Fin))
52	49, 51	mpbird 167	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝒫 1o ∈ Fin)
53		f1finf1o 7216	. . . . . 6 ⊢ ((𝑇 ≈ 𝒫 1o ∧ 𝒫 1o ∈ Fin) → (𝐺:𝑇–1-1→𝒫 1o ↔ 𝐺:𝑇–1-1-onto→𝒫 1o))
54	47, 52, 53	syl2anc 411	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 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 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 𝐺:𝑇–1-1-onto→𝒫 1o)
56	55, 20	sylib 122	. . 3 ⊢ (((𝑁 ∈ ω ∧ 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 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → ran 𝐺 = 𝒫 1o)
58	38, 57	impbida 600	1 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2o ∧ EXMID)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  ∅c0 3507  𝒫 cpw 3668  {csn 3688  ∪ ciun 3990   class class class wbr 4108  EXMIDwem 4306  ωcom 4711   × cxp 4746  ran crn 4749  Rel wrel 4753  –1-1→wf1 5348  –1-1-onto→wf1o 5350  1_oc1o 6639  2_oc2o 6640   ≈ cen 6972   ≼ cdom 6973  Fincfn 6974

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-exmid 4307  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977

This theorem is referenced by: (None)