Theorem isomninnlem 16309

Description: Lemma for isomninn 16310. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)

Hypothesis

Ref	Expression
isomninnlem.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)

Assertion

Ref	Expression
isomninnlem	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))

Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝐺,𝑥   𝑓,𝑉,𝑥

Proof of Theorem isomninnlem

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isomnimap 7272	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)))
2		fveq1 5602	. . . . . . . . 9 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → (𝑔‘𝑥) = ((◡𝐺 ∘ 𝑓)‘𝑥))
3	2	eqeq1d 2218	. . . . . . . 8 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → ((𝑔‘𝑥) = ∅ ↔ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅))
4	3	rexbidv 2511	. . . . . . 7 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → (∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅))
5	2	eqeq1d 2218	. . . . . . . 8 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → ((𝑔‘𝑥) = 1o ↔ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o))
6	5	ralbidv 2510	. . . . . . 7 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o))
7	4, 6	orbi12d 797	. . . . . 6 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → ((∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o) ↔ (∃𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o)))
8		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
9		isomninnlem.g	. . . . . . . . 9 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
10	9	012of 16268	. . . . . . . 8 ⊢ (◡𝐺 ↾ {0, 1}):{0, 1}⟶2o
11		elmapi 6787	. . . . . . . . 9 ⊢ (𝑓 ∈ ({0, 1} ↑𝑚 𝐴) → 𝑓:𝐴⟶{0, 1})
12	11	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → 𝑓:𝐴⟶{0, 1})
13		fco2 5466	. . . . . . . 8 ⊢ (((◡𝐺 ↾ {0, 1}):{0, 1}⟶2o ∧ 𝑓:𝐴⟶{0, 1}) → (◡𝐺 ∘ 𝑓):𝐴⟶2o)
14	10, 12, 13	sylancr 414	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (◡𝐺 ∘ 𝑓):𝐴⟶2o)
15		2onn 6637	. . . . . . . . 9 ⊢ 2o ∈ ω
16	15	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → 2o ∈ ω)
17		simpll 527	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → 𝐴 ∈ 𝑉)
18	16, 17	elmapd 6779	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → ((◡𝐺 ∘ 𝑓) ∈ (2o ↑𝑚 𝐴) ↔ (◡𝐺 ∘ 𝑓):𝐴⟶2o))
19	14, 18	mpbird 167	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (◡𝐺 ∘ 𝑓) ∈ (2o ↑𝑚 𝐴))
20	7, 8, 19	rspcdva 2892	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o))
21		nfv 1554	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
22		nfcv 2352	. . . . . . . . . 10 ⊢ Ⅎ𝑥(2o ↑𝑚 𝐴)
23		nfre1 2553	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅
24		nfra1 2541	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o
25	23, 24	nfor 1600	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)
26	22, 25	nfralxy 2548	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)
27	21, 26	nfan 1591	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
28		nfv 1554	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)
29	27, 28	nfan 1591	. . . . . . 7 ⊢ Ⅎ𝑥((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴))
30	9	frechashgf1o 10617	. . . . . . . . . . 11 ⊢ 𝐺:ω–1-1-onto→ℕ0
31		0nn0 9352	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
32		1nn0 9353	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ0
33		prssi 3805	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
34	31, 32, 33	mp2an 426	. . . . . . . . . . . 12 ⊢ {0, 1} ⊆ ℕ0
35	11	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑓:𝐴⟶{0, 1})
36		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
37	35, 36	ffvelcdmd 5744	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ {0, 1})
38	34, 37	sselid 3202	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ℕ0)
39		f1ocnvfv2 5875	. . . . . . . . . . 11 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ (𝑓‘𝑥) ∈ ℕ0) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝑓‘𝑥))
40	30, 38, 39	sylancr 414	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝑓‘𝑥))
41	40	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝑓‘𝑥))
42		fvco3 5678	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶{0, 1} ∧ 𝑥 ∈ 𝐴) → ((◡𝐺 ∘ 𝑓)‘𝑥) = (◡𝐺‘(𝑓‘𝑥)))
43	35, 42	sylancom 420	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺 ∘ 𝑓)‘𝑥) = (◡𝐺‘(𝑓‘𝑥)))
44	43	eqeq1d 2218	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((◡𝐺 ∘ 𝑓)‘𝑥) = ∅ ↔ (◡𝐺‘(𝑓‘𝑥)) = ∅))
45	44	biimpa 296	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅) → (◡𝐺‘(𝑓‘𝑥)) = ∅)
46	45	fveq2d 5607	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝐺‘∅))
47		0zd 9426	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
48	47, 9	frec2uz0d 10588	. . . . . . . . . . 11 ⊢ (⊤ → (𝐺‘∅) = 0)
49	48	mptru 1384	. . . . . . . . . 10 ⊢ (𝐺‘∅) = 0
50	46, 49	eqtrdi 2258	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = 0)
51	41, 50	eqtr3d 2244	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅) → (𝑓‘𝑥) = 0)
52	51	exp31 364	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (𝑥 ∈ 𝐴 → (((◡𝐺 ∘ 𝑓)‘𝑥) = ∅ → (𝑓‘𝑥) = 0)))
53	29, 52	reximdai 2608	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅ → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
54	40	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝑓‘𝑥))
55	43	adantr 276	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → ((◡𝐺 ∘ 𝑓)‘𝑥) = (◡𝐺‘(𝑓‘𝑥)))
56		simpr 110	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o)
57	55, 56	eqtr3d 2244	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → (◡𝐺‘(𝑓‘𝑥)) = 1o)
58	57	fveq2d 5607	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝐺‘1o))
59		df-1o 6532	. . . . . . . . . . . 12 ⊢ 1o = suc ∅
60	59	fveq2i 5606	. . . . . . . . . . 11 ⊢ (𝐺‘1o) = (𝐺‘suc ∅)
61		peano1 4663	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
62	61	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ ω)
63	47, 9, 62	frec2uzsucd 10590	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
64	63	mptru 1384	. . . . . . . . . . 11 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
65	49	oveq1i 5984	. . . . . . . . . . . 12 ⊢ ((𝐺‘∅) + 1) = (0 + 1)
66		0p1e1 9192	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
67	65, 66	eqtri 2230	. . . . . . . . . . 11 ⊢ ((𝐺‘∅) + 1) = 1
68	60, 64, 67	3eqtri 2234	. . . . . . . . . 10 ⊢ (𝐺‘1o) = 1
69	58, 68	eqtrdi 2258	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = 1)
70	54, 69	eqtr3d 2244	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → (𝑓‘𝑥) = 1)
71	70	ex 115	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((◡𝐺 ∘ 𝑓)‘𝑥) = 1o → (𝑓‘𝑥) = 1))
72	29, 71	ralimdaa 2576	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (∀𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
73	53, 72	orim12d 790	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → ((∃𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))
74	20, 73	mpd 13	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
75	74	ralrimiva 2583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)) → ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
76		fveq1 5602	. . . . . . . . 9 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → (𝑓‘𝑥) = ((𝐺 ∘ 𝑔)‘𝑥))
77	76	eqeq1d 2218	. . . . . . . 8 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → ((𝑓‘𝑥) = 0 ↔ ((𝐺 ∘ 𝑔)‘𝑥) = 0))
78	77	rexbidv 2511	. . . . . . 7 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 0))
79	76	eqeq1d 2218	. . . . . . . 8 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → ((𝑓‘𝑥) = 1 ↔ ((𝐺 ∘ 𝑔)‘𝑥) = 1))
80	79	ralbidv 2510	. . . . . . 7 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 ↔ ∀𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 1))
81	78, 80	orbi12d 797	. . . . . 6 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1) ↔ (∃𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 1)))
82		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
83	9	2o01f 16269	. . . . . . . 8 ⊢ (𝐺 ↾ 2o):2o⟶{0, 1}
84		elmapi 6787	. . . . . . . . 9 ⊢ (𝑔 ∈ (2o ↑𝑚 𝐴) → 𝑔:𝐴⟶2o)
85	84	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → 𝑔:𝐴⟶2o)
86		fco2 5466	. . . . . . . 8 ⊢ (((𝐺 ↾ 2o):2o⟶{0, 1} ∧ 𝑔:𝐴⟶2o) → (𝐺 ∘ 𝑔):𝐴⟶{0, 1})
87	83, 85, 86	sylancr 414	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (𝐺 ∘ 𝑔):𝐴⟶{0, 1})
88		prexg 4274	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ∈ V)
89	31, 32, 88	mp2an 426	. . . . . . . . 9 ⊢ {0, 1} ∈ V
90	89	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → {0, 1} ∈ V)
91		simpll 527	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → 𝐴 ∈ 𝑉)
92	90, 91	elmapd 6779	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → ((𝐺 ∘ 𝑔) ∈ ({0, 1} ↑𝑚 𝐴) ↔ (𝐺 ∘ 𝑔):𝐴⟶{0, 1}))
93	87, 92	mpbird 167	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (𝐺 ∘ 𝑔) ∈ ({0, 1} ↑𝑚 𝐴))
94	81, 82, 93	rspcdva 2892	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 1))
95		nfcv 2352	. . . . . . . . . 10 ⊢ Ⅎ𝑥({0, 1} ↑𝑚 𝐴)
96		nfre1 2553	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0
97		nfra1 2541	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1
98	96, 97	nfor 1600	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)
99	95, 98	nfralxy 2548	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)
100	21, 99	nfan 1591	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
101		nfv 1554	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑔 ∈ (2o ↑𝑚 𝐴)
102	100, 101	nfan 1591	. . . . . . 7 ⊢ Ⅎ𝑥((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴))
103	84	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑔:𝐴⟶2o)
104		omelon 4678	. . . . . . . . . . . . . . . 16 ⊢ ω ∈ On
105	104	onelssi 4497	. . . . . . . . . . . . . . 15 ⊢ (2o ∈ ω → 2o ⊆ ω)
106	15, 105	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2o ⊆ ω
107	106	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 2o ⊆ ω)
108	103, 107	fssd 5462	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑔:𝐴⟶ω)
109		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
110	108, 109	ffvelcdmd 5744	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ω)
111		f1ocnvfv1 5874	. . . . . . . . . . 11 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ (𝑔‘𝑥) ∈ ω) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (𝑔‘𝑥))
112	30, 110, 111	sylancr 414	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (𝑔‘𝑥))
113	112	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 0) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (𝑔‘𝑥))
114		fvco3 5678	. . . . . . . . . . . . . 14 ⊢ ((𝑔:𝐴⟶2o ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝑔)‘𝑥) = (𝐺‘(𝑔‘𝑥)))
115	103, 114	sylancom 420	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝑔)‘𝑥) = (𝐺‘(𝑔‘𝑥)))
116	115	eqeq1d 2218	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝑔)‘𝑥) = 0 ↔ (𝐺‘(𝑔‘𝑥)) = 0))
117	116	biimpa 296	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 0) → (𝐺‘(𝑔‘𝑥)) = 0)
118	117	fveq2d 5607	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 0) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (◡𝐺‘0))
119		f1ocnvfv 5876	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅))
120	30, 61, 119	mp2an 426	. . . . . . . . . . 11 ⊢ ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅)
121	49, 120	ax-mp 5	. . . . . . . . . 10 ⊢ (◡𝐺‘0) = ∅
122	118, 121	eqtrdi 2258	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 0) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = ∅)
123	113, 122	eqtr3d 2244	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 0) → (𝑔‘𝑥) = ∅)
124	123	exp31 364	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (𝑥 ∈ 𝐴 → (((𝐺 ∘ 𝑔)‘𝑥) = 0 → (𝑔‘𝑥) = ∅)))
125	102, 124	reximdai 2608	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 0 → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅))
126	112	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 1) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (𝑔‘𝑥))
127	115	eqeq1d 2218	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝑔)‘𝑥) = 1 ↔ (𝐺‘(𝑔‘𝑥)) = 1))
128	127	biimpa 296	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 1) → (𝐺‘(𝑔‘𝑥)) = 1)
129	128	fveq2d 5607	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 1) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (◡𝐺‘1))
130		1onn 6636	. . . . . . . . . . . 12 ⊢ 1o ∈ ω
131		f1ocnvfv 5876	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ 1o ∈ ω) → ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o))
132	30, 130, 131	mp2an 426	. . . . . . . . . . 11 ⊢ ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o)
133	68, 132	ax-mp 5	. . . . . . . . . 10 ⊢ (◡𝐺‘1) = 1o
134	129, 133	eqtrdi 2258	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 1) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = 1o)
135	126, 134	eqtr3d 2244	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ ((𝐺 ∘ 𝑔)‘𝑥) = 1) → (𝑔‘𝑥) = 1o)
136	135	ex 115	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝑔)‘𝑥) = 1 → (𝑔‘𝑥) = 1o))
137	102, 136	ralimdaa 2576	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (∀𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 1 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
138	125, 137	orim12d 790	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → ((∃𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 1) → (∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)))
139	94, 138	mpd 13	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
140	139	ralrimiva 2583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) → ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
141	75, 140	impbida 598	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o) ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))
142	1, 141	bitrd 188	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 712   = wceq 1375  ⊤wtru 1376   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  Vcvv 2779   ⊆ wss 3177  ∅c0 3471  {cpr 3647   ↦ cmpt 4124  suc csuc 4433  ωcom 4659  ^◡ccnv 4695   ↾ cres 4698   ∘ ccom 4700  ⟶wf 5290  –1-1-onto→wf1o 5293  ‘cfv 5294  (class class class)co 5974  freccfrec 6506  1_oc1o 6525  2_oc2o 6526   ↑_𝑚 cmap 6765  Omnicomni 7269  0cc0 7967  1c1 7968   + caddc 7970  ℕ₀cn0 9337  ℤcz 9414

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-recs 6421  df-frec 6507  df-1o 6532  df-2o 6533  df-map 6767  df-omni 7270  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-n0 9338  df-z 9415  df-uz 9691

This theorem is referenced by:  isomninn  16310