Theorem isomninnlem 13400

Description: Lemma for isomninn 13401. 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 7017	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)))
2		fveq1 5428	. . . . . . . . 9 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → (𝑔‘𝑥) = ((◡𝐺 ∘ 𝑓)‘𝑥))
3	2	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → ((𝑔‘𝑥) = ∅ ↔ ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅))
4	3	rexbidv 2439	. . . . . . 7 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → (∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = ∅))
5	2	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → ((𝑔‘𝑥) = 1o ↔ ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o))
6	5	ralbidv 2438	. . . . . . 7 ⊢ (𝑔 = (◡𝐺 ∘ 𝑓) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((◡𝐺 ∘ 𝑓)‘𝑥) = 1o))
7	4, 6	orbi12d 783	. . . . . 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 520	. . . . . 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 13363	. . . . . . . 8 ⊢ (◡𝐺 ↾ {0, 1}):{0, 1}⟶2o
11		elmapi 6572	. . . . . . . . 9 ⊢ (𝑓 ∈ ({0, 1} ↑𝑚 𝐴) → 𝑓:𝐴⟶{0, 1})
12	11	adantl 275	. . . . . . . 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 5297	. . . . . . . 8 ⊢ (((◡𝐺 ↾ {0, 1}):{0, 1}⟶2o ∧ 𝑓:𝐴⟶{0, 1}) → (◡𝐺 ∘ 𝑓):𝐴⟶2o)
14	10, 12, 13	sylancr 411	. . . . . . 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 6425	. . . . . . . . 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 519	. . . . . . . 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 6564	. . . . . . 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 166	. . . . . 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 2798	. . . . 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 1509	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
22		nfcv 2282	. . . . . . . . . 10 ⊢ Ⅎ𝑥(2o ↑𝑚 𝐴)
23		nfre1 2479	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅
24		nfra1 2469	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o
25	23, 24	nfor 1554	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)
26	22, 25	nfralxy 2474	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o)
27	21, 26	nfan 1545	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
28		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 ∈ ({0, 1} ↑𝑚 𝐴)
29	27, 28	nfan 1545	. . . . . . 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 10232	. . . . . . . . . . 11 ⊢ 𝐺:ω–1-1-onto→ℕ0
31		0nn0 9016	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
32		1nn0 9017	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ0
33		prssi 3686	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
34	31, 32, 33	mp2an 423	. . . . . . . . . . . 12 ⊢ {0, 1} ⊆ ℕ0
35	11	ad2antlr 481	. . . . . . . . . . . . 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 109	. . . . . . . . . . . . 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	ffvelrnd 5564	. . . . . . . . . . . 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	sseldi 3100	. . . . . . . . . . 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 5687	. . . . . . . . . . 11 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ (𝑓‘𝑥) ∈ ℕ0) → (𝐺‘(◡𝐺‘(𝑓‘𝑥))) = (𝑓‘𝑥))
40	30, 38, 39	sylancr 411	. . . . . . . . . 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 274	. . . . . . . . 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 5500	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶{0, 1} ∧ 𝑥 ∈ 𝐴) → ((◡𝐺 ∘ 𝑓)‘𝑥) = (◡𝐺‘(𝑓‘𝑥)))
43	35, 42	sylancom 417	. . . . . . . . . . . . 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 2149	. . . . . . . . . . . 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 294	. . . . . . . . . . 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 5433	. . . . . . . . . 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 9090	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
48	47, 9	frec2uz0d 10203	. . . . . . . . . . 11 ⊢ (⊤ → (𝐺‘∅) = 0)
49	48	mptru 1341	. . . . . . . . . 10 ⊢ (𝐺‘∅) = 0
50	46, 49	eqtrdi 2189	. . . . . . . . 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 2175	. . . . . . . 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 362	. . . . . . 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 2533	. . . . . 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 274	. . . . . . . . 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 274	. . . . . . . . . . . 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 109	. . . . . . . . . . . 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 2175	. . . . . . . . . . 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 5433	. . . . . . . . . 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 6321	. . . . . . . . . . . 12 ⊢ 1o = suc ∅
60	59	fveq2i 5432	. . . . . . . . . . 11 ⊢ (𝐺‘1o) = (𝐺‘suc ∅)
61		peano1 4516	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
62	61	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ ω)
63	47, 9, 62	frec2uzsucd 10205	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
64	63	mptru 1341	. . . . . . . . . . 11 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
65	49	oveq1i 5792	. . . . . . . . . . . 12 ⊢ ((𝐺‘∅) + 1) = (0 + 1)
66		0p1e1 8858	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
67	65, 66	eqtri 2161	. . . . . . . . . . 11 ⊢ ((𝐺‘∅) + 1) = 1
68	60, 64, 67	3eqtri 2165	. . . . . . . . . 10 ⊢ (𝐺‘1o) = 1
69	58, 68	eqtrdi 2189	. . . . . . . . 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 2175	. . . . . . . 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 114	. . . . . . 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 2501	. . . . . 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 776	. . . . 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 2508	. . 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 5428	. . . . . . . . 9 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → (𝑓‘𝑥) = ((𝐺 ∘ 𝑔)‘𝑥))
77	76	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → ((𝑓‘𝑥) = 0 ↔ ((𝐺 ∘ 𝑔)‘𝑥) = 0))
78	77	rexbidv 2439	. . . . . . 7 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 0))
79	76	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → ((𝑓‘𝑥) = 1 ↔ ((𝐺 ∘ 𝑔)‘𝑥) = 1))
80	79	ralbidv 2438	. . . . . . 7 ⊢ (𝑓 = (𝐺 ∘ 𝑔) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 ↔ ∀𝑥 ∈ 𝐴 ((𝐺 ∘ 𝑔)‘𝑥) = 1))
81	78, 80	orbi12d 783	. . . . . 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 520	. . . . . 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 13364	. . . . . . . 8 ⊢ (𝐺 ↾ 2o):2o⟶{0, 1}
84		elmapi 6572	. . . . . . . . 9 ⊢ (𝑔 ∈ (2o ↑𝑚 𝐴) → 𝑔:𝐴⟶2o)
85	84	adantl 275	. . . . . . . 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 5297	. . . . . . . 8 ⊢ (((𝐺 ↾ 2o):2o⟶{0, 1} ∧ 𝑔:𝐴⟶2o) → (𝐺 ∘ 𝑔):𝐴⟶{0, 1})
87	83, 85, 86	sylancr 411	. . . . . . 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 4141	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ∈ V)
89	31, 32, 88	mp2an 423	. . . . . . . . 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 519	. . . . . . . 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 6564	. . . . . . 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 166	. . . . . 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 2798	. . . . 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 2282	. . . . . . . . . 10 ⊢ Ⅎ𝑥({0, 1} ↑𝑚 𝐴)
96		nfre1 2479	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0
97		nfra1 2469	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1
98	96, 97	nfor 1554	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)
99	95, 98	nfralxy 2474	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)
100	21, 99	nfan 1545	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
101		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑔 ∈ (2o ↑𝑚 𝐴)
102	100, 101	nfan 1545	. . . . . . 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 481	. . . . . . . . . . . . 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 4530	. . . . . . . . . . . . . . . 16 ⊢ ω ∈ On
105	104	onelssi 4359	. . . . . . . . . . . . . . 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 5293	. . . . . . . . . . . 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 109	. . . . . . . . . . . 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	ffvelrnd 5564	. . . . . . . . . . 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 5686	. . . . . . . . . . 11 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ (𝑔‘𝑥) ∈ ω) → (◡𝐺‘(𝐺‘(𝑔‘𝑥))) = (𝑔‘𝑥))
112	30, 110, 111	sylancr 411	. . . . . . . . . 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 274	. . . . . . . . 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 5500	. . . . . . . . . . . . . 14 ⊢ ((𝑔:𝐴⟶2o ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝑔)‘𝑥) = (𝐺‘(𝑔‘𝑥)))
115	103, 114	sylancom 417	. . . . . . . . . . . . 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 2149	. . . . . . . . . . . 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 294	. . . . . . . . . . 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 5433	. . . . . . . . . 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 5688	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅))
120	30, 61, 119	mp2an 423	. . . . . . . . . . 11 ⊢ ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅)
121	49, 120	ax-mp 5	. . . . . . . . . 10 ⊢ (◡𝐺‘0) = ∅
122	118, 121	eqtrdi 2189	. . . . . . . . 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 2175	. . . . . . . 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 362	. . . . . . 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 2533	. . . . . 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 274	. . . . . . . . 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 2149	. . . . . . . . . . . 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 294	. . . . . . . . . . 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 5433	. . . . . . . . . 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 6424	. . . . . . . . . . . 12 ⊢ 1o ∈ ω
131		f1ocnvfv 5688	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ 1o ∈ ω) → ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o))
132	30, 130, 131	mp2an 423	. . . . . . . . . . 11 ⊢ ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o)
133	68, 132	ax-mp 5	. . . . . . . . . 10 ⊢ (◡𝐺‘1) = 1o
134	129, 133	eqtrdi 2189	. . . . . . . . 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 2175	. . . . . . . 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 114	. . . . . . 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 2501	. . . . . 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 776	. . . . 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 2508	. . 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 586	. 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 187	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 103   ↔ wb 104   ∨ wo 698   = wceq 1332  ⊤wtru 1333   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ⊆ wss 3076  ∅c0 3368  {cpr 3533   ↦ cmpt 3997  suc csuc 4295  ωcom 4512  ^◡ccnv 4546   ↾ cres 4549   ∘ ccom 4551  ⟶wf 5127  –1-1-onto→wf1o 5130  ‘cfv 5131  (class class class)co 5782  freccfrec 6295  1_oc1o 6314  2_oc2o 6315   ↑_𝑚 cmap 6550  Omnicomni 7012  0cc0 7644  1c1 7645   + caddc 7647  ℕ₀cn0 9001  ℤcz 9078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-recs 6210  df-frec 6296  df-1o 6321  df-2o 6322  df-map 6552  df-omni 7014  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351

This theorem is referenced by:  isomninn  13401