Theorem pwf1oexmid 16621

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 16620	. . . . 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 7445	. . . . . 6 ⊢ 2o ≼ 𝒫 1o
5		iunxpconst 4786	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) = (𝑁 × 1o)
6		df1o2 6596	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
7	6	xpeq2i 4746	. . . . . . . . . . . 12 ⊢ (𝑁 × 1o) = (𝑁 × {∅})
8	1, 5, 7	3eqtri 2256	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑁 × {∅})
9		peano1 4692	. . . . . . . . . . . 12 ⊢ ∅ ∈ ω
10		xpsneng 7006	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ ∅ ∈ ω) → (𝑁 × {∅}) ≈ 𝑁)
11	9, 10	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑁 × {∅}) ≈ 𝑁)
12	8, 11	eqbrtrid 4123	. . . . . . . . . 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 6957	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑁 ≈ 𝑇)
15		relen 6913	. . . . . . . . . 10 ⊢ Rel ≈
16		brrelex1 4765	. . . . . . . . . 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 5592	. . . . . . . . . 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 6930	. . . . . . . . 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 6958	. . . . . . . 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 6957	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝒫 1o ≈ 𝑁)
27		domentr 6965	. . . . . 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 6689	. . . . . . 7 ⊢ 2o ∈ ω
30		nndomo 7050	. . . . . . 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 3244	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝑁 = 2o)
35	26, 34	breqtrd 4114	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ ran 𝐺 = 𝒫 1o) → 𝒫 1o ≈ 2o)
36		exmidpw 7100	. . . 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 4114	. . . . . . 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 6957	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ (𝑁 = 2o ∧ EXMID)) → 2o ≈ 𝒫 1o)
46		entr 6958	. . . . . . 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 7059	. . . . . . . 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 7060	. . . . . . . 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 7146	. . . . . 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 1397   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  ∅c0 3494  𝒫 cpw 3652  {csn 3669  ∪ ciun 3970   class class class wbr 4088  EXMIDwem 4284  ωcom 4688   × cxp 4723  ran crn 4726  Rel wrel 4730  –1-1→wf1 5323  –1-1-onto→wf1o 5325  1_oc1o 6575  2_oc2o 6576   ≈ cen 6907   ≼ cdom 6908  Fincfn 6909

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-exmid 4285  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912

This theorem is referenced by: (None)