Theorem diophrw 43518

Description: Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)

Assertion

Ref	Expression
diophrw	⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)} = {𝑎 ∣ ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)})

Distinct variable groups:   𝑆,𝑎,𝑏,𝑐,𝑑   𝑇,𝑎,𝑏,𝑐,𝑑   𝑀,𝑎,𝑏,𝑐,𝑑   𝑂,𝑎,𝑏,𝑐,𝑑   𝑃,𝑏,𝑐,𝑑

Allowed substitution hint:   𝑃(𝑎)

Proof of Theorem diophrw

Step	Hyp	Ref	Expression
1		simpr 489	. . . . . . . . 9 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) → 𝑏 ∈ (ℕ0 ↑m 𝑆))
2		nn0ex 12514	. . . . . . . . . 10 ⊢ ℕ0 ∈ V
3		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → 𝑆 ∈ V)
4	3	adantr 485	. . . . . . . . . 10 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) → 𝑆 ∈ V)
5		elmapg 8832	. . . . . . . . . 10 ⊢ ((ℕ0 ∈ V ∧ 𝑆 ∈ V) → (𝑏 ∈ (ℕ0 ↑m 𝑆) ↔ 𝑏:𝑆⟶ℕ0))
6	2, 4, 5	sylancr 598	. . . . . . . . 9 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) → (𝑏 ∈ (ℕ0 ↑m 𝑆) ↔ 𝑏:𝑆⟶ℕ0))
7	1, 6	mpbid 235	. . . . . . . 8 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) → 𝑏:𝑆⟶ℕ0)
8	7	adantr 485	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑏:𝑆⟶ℕ0)
9		simp2 1155	. . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → 𝑀:𝑇–1-1→𝑆)
10		f1f 6774	. . . . . . . . 9 ⊢ (𝑀:𝑇–1-1→𝑆 → 𝑀:𝑇⟶𝑆)
11	9, 10	syl 18	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → 𝑀:𝑇⟶𝑆)
12	11	ad2antrr 738	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑀:𝑇⟶𝑆)
13		fco 6730	. . . . . . 7 ⊢ ((𝑏:𝑆⟶ℕ0 ∧ 𝑀:𝑇⟶𝑆) → (𝑏 ∘ 𝑀):𝑇⟶ℕ0)
14	8, 12, 13	syl2anc 595	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (𝑏 ∘ 𝑀):𝑇⟶ℕ0)
15		f1dmex 7950	. . . . . . . . 9 ⊢ ((𝑀:𝑇–1-1→𝑆 ∧ 𝑆 ∈ V) → 𝑇 ∈ V)
16	9, 3, 15	syl2anc 595	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → 𝑇 ∈ V)
17	16	ad2antrr 738	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑇 ∈ V)
18		elmapg 8832	. . . . . . 7 ⊢ ((ℕ0 ∈ V ∧ 𝑇 ∈ V) → ((𝑏 ∘ 𝑀) ∈ (ℕ0 ↑m 𝑇) ↔ (𝑏 ∘ 𝑀):𝑇⟶ℕ0))
19	2, 17, 18	sylancr 598	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → ((𝑏 ∘ 𝑀) ∈ (ℕ0 ↑m 𝑇) ↔ (𝑏 ∘ 𝑀):𝑇⟶ℕ0))
20	14, 19	mpbird 260	. . . . 5 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (𝑏 ∘ 𝑀) ∈ (ℕ0 ↑m 𝑇))
21		simprl 782	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑎 = (𝑏 ↾ 𝑂))
22		resco 6251	. . . . . . 7 ⊢ ((𝑏 ∘ 𝑀) ↾ 𝑂) = (𝑏 ∘ (𝑀 ↾ 𝑂))
23		simpll3 1233	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (𝑀 ↾ 𝑂) = ( I ↾ 𝑂))
24	23	coeq2d 5848	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (𝑏 ∘ (𝑀 ↾ 𝑂)) = (𝑏 ∘ ( I ↾ 𝑂)))
25		coires1 6266	. . . . . . . 8 ⊢ (𝑏 ∘ ( I ↾ 𝑂)) = (𝑏 ↾ 𝑂)
26	24, 25	eqtrdi 2814	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (𝑏 ∘ (𝑀 ↾ 𝑂)) = (𝑏 ↾ 𝑂))
27	22, 26	eqtrid 2810	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → ((𝑏 ∘ 𝑀) ↾ 𝑂) = (𝑏 ↾ 𝑂))
28	21, 27	eqtr4d 2801	. . . . 5 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑎 = ((𝑏 ∘ 𝑀) ↾ 𝑂))
29		simpll1 1231	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑆 ∈ V)
30		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑆 → (ℕ0 ↑m 𝑎) = (ℕ0 ↑m 𝑆))
31		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑆 → (ℤ ↑m 𝑎) = (ℤ ↑m 𝑆))
32	30, 31	sseq12d 3970	. . . . . . . . . 10 ⊢ (𝑎 = 𝑆 → ((ℕ0 ↑m 𝑎) ⊆ (ℤ ↑m 𝑎) ↔ (ℕ0 ↑m 𝑆) ⊆ (ℤ ↑m 𝑆)))
33		zex 12604	. . . . . . . . . . 11 ⊢ ℤ ∈ V
34		nn0ssz 12618	. . . . . . . . . . 11 ⊢ ℕ0 ⊆ ℤ
35		mapss 8883	. . . . . . . . . . 11 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m 𝑎) ⊆ (ℤ ↑m 𝑎))
36	33, 34, 35	mp2an 704	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝑎) ⊆ (ℤ ↑m 𝑎)
37	32, 36	vtoclg 3522	. . . . . . . . 9 ⊢ (𝑆 ∈ V → (ℕ0 ↑m 𝑆) ⊆ (ℤ ↑m 𝑆))
38	29, 37	syl 18	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (ℕ0 ↑m 𝑆) ⊆ (ℤ ↑m 𝑆))
39		simplr 780	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑏 ∈ (ℕ0 ↑m 𝑆))
40	38, 39	sseldd 3938	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → 𝑏 ∈ (ℤ ↑m 𝑆))
41		coeq1 5843	. . . . . . . . 9 ⊢ (𝑑 = 𝑏 → (𝑑 ∘ 𝑀) = (𝑏 ∘ 𝑀))
42	41	fveq2d 6885	. . . . . . . 8 ⊢ (𝑑 = 𝑏 → (𝑃‘(𝑑 ∘ 𝑀)) = (𝑃‘(𝑏 ∘ 𝑀)))
43		eqid 2763	. . . . . . . 8 ⊢ (𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀))) = (𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))
44		fvex 6894	. . . . . . . 8 ⊢ (𝑃‘(𝑏 ∘ 𝑀)) ∈ V
45	42, 43, 44	fvmpt 6989	. . . . . . 7 ⊢ (𝑏 ∈ (ℤ ↑m 𝑆) → ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = (𝑃‘(𝑏 ∘ 𝑀)))
46	40, 45	syl 18	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = (𝑃‘(𝑏 ∘ 𝑀)))
47		simprr 784	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)
48	46, 47	eqtr3d 2800	. . . . 5 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → (𝑃‘(𝑏 ∘ 𝑀)) = 0)
49		reseq1 5972	. . . . . . . 8 ⊢ (𝑐 = (𝑏 ∘ 𝑀) → (𝑐 ↾ 𝑂) = ((𝑏 ∘ 𝑀) ↾ 𝑂))
50	49	eqeq2d 2774	. . . . . . 7 ⊢ (𝑐 = (𝑏 ∘ 𝑀) → (𝑎 = (𝑐 ↾ 𝑂) ↔ 𝑎 = ((𝑏 ∘ 𝑀) ↾ 𝑂)))
51		fveqeq2 6890	. . . . . . 7 ⊢ (𝑐 = (𝑏 ∘ 𝑀) → ((𝑃‘𝑐) = 0 ↔ (𝑃‘(𝑏 ∘ 𝑀)) = 0))
52	50, 51	anbi12d 643	. . . . . 6 ⊢ (𝑐 = (𝑏 ∘ 𝑀) → ((𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0) ↔ (𝑎 = ((𝑏 ∘ 𝑀) ↾ 𝑂) ∧ (𝑃‘(𝑏 ∘ 𝑀)) = 0)))
53	52	rspcev 3581	. . . . 5 ⊢ (((𝑏 ∘ 𝑀) ∈ (ℕ0 ↑m 𝑇) ∧ (𝑎 = ((𝑏 ∘ 𝑀) ↾ 𝑂) ∧ (𝑃‘(𝑏 ∘ 𝑀)) = 0)) → ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0))
54	20, 28, 48, 53	syl12anc 849	. . . 4 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑏 ∈ (ℕ0 ↑m 𝑆)) ∧ (𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)) → ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0))
55	54	rexlimdva2 3168	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → (∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0) → ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)))
56		simpr 489	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑐 ∈ (ℕ0 ↑m 𝑇))
57	16	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑇 ∈ V)
58		elmapg 8832	. . . . . . . . . . . 12 ⊢ ((ℕ0 ∈ V ∧ 𝑇 ∈ V) → (𝑐 ∈ (ℕ0 ↑m 𝑇) ↔ 𝑐:𝑇⟶ℕ0))
59	2, 57, 58	sylancr 598	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑐 ∈ (ℕ0 ↑m 𝑇) ↔ 𝑐:𝑇⟶ℕ0))
60	56, 59	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑐:𝑇⟶ℕ0)
61	60	adantr 485	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → 𝑐:𝑇⟶ℕ0)
62	9	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → 𝑀:𝑇–1-1→𝑆)
63		f1cnv 6845	. . . . . . . . . 10 ⊢ (𝑀:𝑇–1-1→𝑆 → ◡𝑀:ran 𝑀–1-1-onto→𝑇)
64		f1of 6820	. . . . . . . . . 10 ⊢ (◡𝑀:ran 𝑀–1-1-onto→𝑇 → ◡𝑀:ran 𝑀⟶𝑇)
65	62, 63, 64	3syl 19	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ◡𝑀:ran 𝑀⟶𝑇)
66		fco 6730	. . . . . . . . 9 ⊢ ((𝑐:𝑇⟶ℕ0 ∧ ◡𝑀:ran 𝑀⟶𝑇) → (𝑐 ∘ ◡𝑀):ran 𝑀⟶ℕ0)
67	61, 65, 66	syl2anc 595	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑐 ∘ ◡𝑀):ran 𝑀⟶ℕ0)
68		c0ex 11204	. . . . . . . . . 10 ⊢ 0 ∈ V
69	68	fconst 6764	. . . . . . . . 9 ⊢ ((𝑆 ∖ ran 𝑀) × {0}):(𝑆 ∖ ran 𝑀)⟶{0}
70	69	a1i 11	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑆 ∖ ran 𝑀) × {0}):(𝑆 ∖ ran 𝑀)⟶{0})
71		disjdif 4433	. . . . . . . . 9 ⊢ (ran 𝑀 ∩ (𝑆 ∖ ran 𝑀)) = ∅
72	71	a1i 11	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (ran 𝑀 ∩ (𝑆 ∖ ran 𝑀)) = ∅)
73		fun 6740	. . . . . . . 8 ⊢ ((((𝑐 ∘ ◡𝑀):ran 𝑀⟶ℕ0 ∧ ((𝑆 ∖ ran 𝑀) × {0}):(𝑆 ∖ ran 𝑀)⟶{0}) ∧ (ran 𝑀 ∩ (𝑆 ∖ ran 𝑀)) = ∅) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):(ran 𝑀 ∪ (𝑆 ∖ ran 𝑀))⟶(ℕ0 ∪ {0}))
74	67, 70, 72, 73	syl21anc 850	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):(ran 𝑀 ∪ (𝑆 ∖ ran 𝑀))⟶(ℕ0 ∪ {0}))
75		frn 6713	. . . . . . . . . . 11 ⊢ (𝑀:𝑇⟶𝑆 → ran 𝑀 ⊆ 𝑆)
76	9, 10, 75	3syl 19	. . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → ran 𝑀 ⊆ 𝑆)
77	76	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ran 𝑀 ⊆ 𝑆)
78		undif 4443	. . . . . . . . 9 ⊢ (ran 𝑀 ⊆ 𝑆 ↔ (ran 𝑀 ∪ (𝑆 ∖ ran 𝑀)) = 𝑆)
79	77, 78	sylib 221	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (ran 𝑀 ∪ (𝑆 ∖ ran 𝑀)) = 𝑆)
80		0nn0 12523	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
81		snssi 4751	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ0 → {0} ⊆ ℕ0)
82	80, 81	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ ℕ0
83		ssequn2 4142	. . . . . . . . . 10 ⊢ ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
84	82, 83	mpbi 233	. . . . . . . . 9 ⊢ (ℕ0 ∪ {0}) = ℕ0
85	84	a1i 11	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (ℕ0 ∪ {0}) = ℕ0)
86	79, 85	feq23d 6700	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):(ran 𝑀 ∪ (𝑆 ∖ ran 𝑀))⟶(ℕ0 ∪ {0}) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℕ0))
87	74, 86	mpbid 235	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℕ0)
88		elmapg 8832	. . . . . . . 8 ⊢ ((ℕ0 ∈ V ∧ 𝑆 ∈ V) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℕ0 ↑m 𝑆) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℕ0))
89	2, 3, 88	sylancr 598	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℕ0 ↑m 𝑆) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℕ0))
90	89	ad2antrr 738	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℕ0 ↑m 𝑆) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℕ0))
91	87, 90	mpbird 260	. . . . 5 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℕ0 ↑m 𝑆))
92		simprl 782	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → 𝑎 = (𝑐 ↾ 𝑂))
93		resundir 5993	. . . . . . . . 9 ⊢ (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂) = (((𝑐 ∘ ◡𝑀) ↾ 𝑂) ∪ (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂))
94		resco 6251	. . . . . . . . . . 11 ⊢ ((𝑐 ∘ ◡𝑀) ↾ 𝑂) = (𝑐 ∘ (◡𝑀 ↾ 𝑂))
95		simpl2 1211	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑀:𝑇–1-1→𝑆)
96		df-f1 6541	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀:𝑇–1-1→𝑆 ↔ (𝑀:𝑇⟶𝑆 ∧ Fun ◡𝑀))
97	96	simprbi 502	. . . . . . . . . . . . . . . 16 ⊢ (𝑀:𝑇–1-1→𝑆 → Fun ◡𝑀)
98		funcnvres 6614	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡𝑀 → ◡(𝑀 ↾ 𝑂) = (◡𝑀 ↾ (𝑀 “ 𝑂)))
99	95, 97, 98	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ◡(𝑀 ↾ 𝑂) = (◡𝑀 ↾ (𝑀 “ 𝑂)))
100		simpl3 1212	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑀 ↾ 𝑂) = ( I ↾ 𝑂))
101	100	cnveqd 5861	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ◡(𝑀 ↾ 𝑂) = ◡( I ↾ 𝑂))
102		df-ima 5674	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 “ 𝑂) = ran (𝑀 ↾ 𝑂)
103	100	rneqd 5928	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ran (𝑀 ↾ 𝑂) = ran ( I ↾ 𝑂))
104		rnresi 6077	. . . . . . . . . . . . . . . . . 18 ⊢ ran ( I ↾ 𝑂) = 𝑂
105	103, 104	eqtrdi 2814	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ran (𝑀 ↾ 𝑂) = 𝑂)
106	102, 105	eqtrid 2810	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑀 “ 𝑂) = 𝑂)
107	106	reseq2d 5978	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (◡𝑀 ↾ (𝑀 “ 𝑂)) = (◡𝑀 ↾ 𝑂))
108	99, 101, 107	3eqtr3d 2806	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ◡( I ↾ 𝑂) = (◡𝑀 ↾ 𝑂))
109		cnvresid 6615	. . . . . . . . . . . . . 14 ⊢ ◡( I ↾ 𝑂) = ( I ↾ 𝑂)
110	108, 109	eqtr3di 2813	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (◡𝑀 ↾ 𝑂) = ( I ↾ 𝑂))
111	110	coeq2d 5848	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑐 ∘ (◡𝑀 ↾ 𝑂)) = (𝑐 ∘ ( I ↾ 𝑂)))
112		coires1 6266	. . . . . . . . . . . 12 ⊢ (𝑐 ∘ ( I ↾ 𝑂)) = (𝑐 ↾ 𝑂)
113	111, 112	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑐 ∘ (◡𝑀 ↾ 𝑂)) = (𝑐 ↾ 𝑂))
114	94, 113	eqtrid 2810	. . . . . . . . . 10 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ((𝑐 ∘ ◡𝑀) ↾ 𝑂) = (𝑐 ↾ 𝑂))
115		dmres 6011	. . . . . . . . . . . 12 ⊢ dom (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = (𝑂 ∩ dom ((𝑆 ∖ ran 𝑀) × {0}))
116	68	snnz 4742	. . . . . . . . . . . . . . 15 ⊢ {0} ≠ ∅
117		dmxp 5919	. . . . . . . . . . . . . . 15 ⊢ ({0} ≠ ∅ → dom ((𝑆 ∖ ran 𝑀) × {0}) = (𝑆 ∖ ran 𝑀))
118	116, 117	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom ((𝑆 ∖ ran 𝑀) × {0}) = (𝑆 ∖ ran 𝑀)
119	118	ineq2i 4170	. . . . . . . . . . . . 13 ⊢ (𝑂 ∩ dom ((𝑆 ∖ ran 𝑀) × {0})) = (𝑂 ∩ (𝑆 ∖ ran 𝑀))
120		inss1 4189	. . . . . . . . . . . . . . 15 ⊢ (𝑂 ∩ 𝑆) ⊆ 𝑂
121	103, 104	eqtr2di 2815	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑂 = ran (𝑀 ↾ 𝑂))
122		resss 6000	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ↾ 𝑂) ⊆ 𝑀
123		rnss 5929	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ↾ 𝑂) ⊆ 𝑀 → ran (𝑀 ↾ 𝑂) ⊆ ran 𝑀)
124	122, 123	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → ran (𝑀 ↾ 𝑂) ⊆ ran 𝑀)
125	121, 124	eqsstrd 3971	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑂 ⊆ ran 𝑀)
126	120, 125	sstrid 3948	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑂 ∩ 𝑆) ⊆ ran 𝑀)
127		inssdif0 4329	. . . . . . . . . . . . . 14 ⊢ ((𝑂 ∩ 𝑆) ⊆ ran 𝑀 ↔ (𝑂 ∩ (𝑆 ∖ ran 𝑀)) = ∅)
128	126, 127	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑂 ∩ (𝑆 ∖ ran 𝑀)) = ∅)
129	119, 128	eqtrid 2810	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑂 ∩ dom ((𝑆 ∖ ran 𝑀) × {0})) = ∅)
130	115, 129	eqtrid 2810	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → dom (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = ∅)
131		relres 6004	. . . . . . . . . . . 12 ⊢ Rel (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂)
132		reldm0 5918	. . . . . . . . . . . 12 ⊢ (Rel (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) → ((((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = ∅ ↔ dom (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = ∅))
133	131, 132	ax-mp 5	. . . . . . . . . . 11 ⊢ ((((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = ∅ ↔ dom (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = ∅)
134	130, 133	sylibr 237	. . . . . . . . . 10 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂) = ∅)
135	114, 134	uneq12d 4123	. . . . . . . . 9 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (((𝑐 ∘ ◡𝑀) ↾ 𝑂) ∪ (((𝑆 ∖ ran 𝑀) × {0}) ↾ 𝑂)) = ((𝑐 ↾ 𝑂) ∪ ∅))
136	93, 135	eqtrid 2810	. . . . . . . 8 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂) = ((𝑐 ↾ 𝑂) ∪ ∅))
137		un0 4351	. . . . . . . 8 ⊢ ((𝑐 ↾ 𝑂) ∪ ∅) = (𝑐 ↾ 𝑂)
138	136, 137	eqtr2di 2815	. . . . . . 7 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → (𝑐 ↾ 𝑂) = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂))
139	138	adantr 485	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑐 ↾ 𝑂) = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂))
140	92, 139	eqtrd 2798	. . . . 5 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → 𝑎 = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂))
141		fss 6722	. . . . . . . . . . . . 13 ⊢ ((𝑐:𝑇⟶ℕ0 ∧ ℕ0 ⊆ ℤ) → 𝑐:𝑇⟶ℤ)
142	60, 34, 141	sylancl 597	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) → 𝑐:𝑇⟶ℤ)
143	142	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → 𝑐:𝑇⟶ℤ)
144		fco 6730	. . . . . . . . . . 11 ⊢ ((𝑐:𝑇⟶ℤ ∧ ◡𝑀:ran 𝑀⟶𝑇) → (𝑐 ∘ ◡𝑀):ran 𝑀⟶ℤ)
145	143, 65, 144	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑐 ∘ ◡𝑀):ran 𝑀⟶ℤ)
146		fun 6740	. . . . . . . . . 10 ⊢ ((((𝑐 ∘ ◡𝑀):ran 𝑀⟶ℤ ∧ ((𝑆 ∖ ran 𝑀) × {0}):(𝑆 ∖ ran 𝑀)⟶{0}) ∧ (ran 𝑀 ∩ (𝑆 ∖ ran 𝑀)) = ∅) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):(ran 𝑀 ∪ (𝑆 ∖ ran 𝑀))⟶(ℤ ∪ {0}))
147	145, 70, 72, 146	syl21anc 850	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):(ran 𝑀 ∪ (𝑆 ∖ ran 𝑀))⟶(ℤ ∪ {0}))
148		0z 12606	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
149		snssi 4751	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
150	148, 149	ax-mp 5	. . . . . . . . . . . 12 ⊢ {0} ⊆ ℤ
151		ssequn2 4142	. . . . . . . . . . . 12 ⊢ ({0} ⊆ ℤ ↔ (ℤ ∪ {0}) = ℤ)
152	150, 151	mpbi 233	. . . . . . . . . . 11 ⊢ (ℤ ∪ {0}) = ℤ
153	152	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (ℤ ∪ {0}) = ℤ)
154	79, 153	feq23d 6700	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):(ran 𝑀 ∪ (𝑆 ∖ ran 𝑀))⟶(ℤ ∪ {0}) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℤ))
155	147, 154	mpbid 235	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℤ)
156		elmapg 8832	. . . . . . . . . 10 ⊢ ((ℤ ∈ V ∧ 𝑆 ∈ V) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℤ ↑m 𝑆) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℤ))
157	33, 3, 156	sylancr 598	. . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℤ ↑m 𝑆) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℤ))
158	157	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℤ ↑m 𝑆) ↔ ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})):𝑆⟶ℤ))
159	155, 158	mpbird 260	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℤ ↑m 𝑆))
160		coeq1 5843	. . . . . . . . 9 ⊢ (𝑑 = ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) → (𝑑 ∘ 𝑀) = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀))
161	160	fveq2d 6885	. . . . . . . 8 ⊢ (𝑑 = ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) → (𝑃‘(𝑑 ∘ 𝑀)) = (𝑃‘(((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀)))
162		fvex 6894	. . . . . . . 8 ⊢ (𝑃‘(((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀)) ∈ V
163	161, 43, 162	fvmpt 6989	. . . . . . 7 ⊢ (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℤ ↑m 𝑆) → ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0}))) = (𝑃‘(((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀)))
164	159, 163	syl 18	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0}))) = (𝑃‘(((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀)))
165		coundir 6249	. . . . . . . 8 ⊢ (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀) = (((𝑐 ∘ ◡𝑀) ∘ 𝑀) ∪ (((𝑆 ∖ ran 𝑀) × {0}) ∘ 𝑀))
166		coass 6267	. . . . . . . . . . 11 ⊢ ((𝑐 ∘ ◡𝑀) ∘ 𝑀) = (𝑐 ∘ (◡𝑀 ∘ 𝑀))
167		f1cocnv1 6851	. . . . . . . . . . . . 13 ⊢ (𝑀:𝑇–1-1→𝑆 → (◡𝑀 ∘ 𝑀) = ( I ↾ 𝑇))
168	167	coeq2d 5848	. . . . . . . . . . . 12 ⊢ (𝑀:𝑇–1-1→𝑆 → (𝑐 ∘ (◡𝑀 ∘ 𝑀)) = (𝑐 ∘ ( I ↾ 𝑇)))
169	62, 168	syl 18	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑐 ∘ (◡𝑀 ∘ 𝑀)) = (𝑐 ∘ ( I ↾ 𝑇)))
170	166, 169	eqtrid 2810	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ◡𝑀) ∘ 𝑀) = (𝑐 ∘ ( I ↾ 𝑇)))
171	118	ineq1i 4169	. . . . . . . . . . . . 13 ⊢ (dom ((𝑆 ∖ ran 𝑀) × {0}) ∩ ran 𝑀) = ((𝑆 ∖ ran 𝑀) ∩ ran 𝑀)
172		incom 4162	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∖ ran 𝑀) ∩ ran 𝑀) = (ran 𝑀 ∩ (𝑆 ∖ ran 𝑀))
173	171, 172, 71	3eqtri 2790	. . . . . . . . . . . 12 ⊢ (dom ((𝑆 ∖ ran 𝑀) × {0}) ∩ ran 𝑀) = ∅
174		coeq0 6257	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∖ ran 𝑀) × {0}) ∘ 𝑀) = ∅ ↔ (dom ((𝑆 ∖ ran 𝑀) × {0}) ∩ ran 𝑀) = ∅)
175	173, 174	mpbir 234	. . . . . . . . . . 11 ⊢ (((𝑆 ∖ ran 𝑀) × {0}) ∘ 𝑀) = ∅
176	175	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑆 ∖ ran 𝑀) × {0}) ∘ 𝑀) = ∅)
177	170, 176	uneq12d 4123	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∘ 𝑀) ∪ (((𝑆 ∖ ran 𝑀) × {0}) ∘ 𝑀)) = ((𝑐 ∘ ( I ↾ 𝑇)) ∪ ∅))
178		un0 4351	. . . . . . . . . 10 ⊢ ((𝑐 ∘ ( I ↾ 𝑇)) ∪ ∅) = (𝑐 ∘ ( I ↾ 𝑇))
179		fcoi1 6752	. . . . . . . . . . 11 ⊢ (𝑐:𝑇⟶ℕ0 → (𝑐 ∘ ( I ↾ 𝑇)) = 𝑐)
180	61, 179	syl 18	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑐 ∘ ( I ↾ 𝑇)) = 𝑐)
181	178, 180	eqtrid 2810	. . . . . . . . 9 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑐 ∘ ( I ↾ 𝑇)) ∪ ∅) = 𝑐)
182	177, 181	eqtrd 2798	. . . . . . . 8 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∘ 𝑀) ∪ (((𝑆 ∖ ran 𝑀) × {0}) ∘ 𝑀)) = 𝑐)
183	165, 182	eqtrid 2810	. . . . . . 7 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀) = 𝑐)
184	183	fveq2d 6885	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑃‘(((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∘ 𝑀)) = (𝑃‘𝑐))
185		simprr 784	. . . . . 6 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → (𝑃‘𝑐) = 0)
186	164, 184, 185	3eqtrd 2802	. . . . 5 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0}))) = 0)
187		reseq1 5972	. . . . . . . 8 ⊢ (𝑏 = ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) → (𝑏 ↾ 𝑂) = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂))
188	187	eqeq2d 2774	. . . . . . 7 ⊢ (𝑏 = ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) → (𝑎 = (𝑏 ↾ 𝑂) ↔ 𝑎 = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂)))
189		fveqeq2 6890	. . . . . . 7 ⊢ (𝑏 = ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) → (((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0 ↔ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0}))) = 0))
190	188, 189	anbi12d 643	. . . . . 6 ⊢ (𝑏 = ((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) → ((𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0) ↔ (𝑎 = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0}))) = 0)))
191	190	rspcev 3581	. . . . 5 ⊢ ((((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ∈ (ℕ0 ↑m 𝑆) ∧ (𝑎 = (((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0})) ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘((𝑐 ∘ ◡𝑀) ∪ ((𝑆 ∖ ran 𝑀) × {0}))) = 0)) → ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0))
192	91, 140, 186, 191	syl12anc 849	. . . 4 ⊢ ((((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) ∧ 𝑐 ∈ (ℕ0 ↑m 𝑇)) ∧ (𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)) → ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0))
193	192	rexlimdva2 3168	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → (∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0) → ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)))
194	55, 193	impbid 215	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → (∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0) ↔ ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)))
195	194	abbidv 2829	1 ⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)} = {𝑎 ∣ ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∃wrex 3089  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4589   ↦ cmpt 5192   I cid 5555   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  Rel wrel 5666  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  0cc0 11104  ℕ₀cn0 12508  ℤcz 12595

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-i2m1 11172  ax-1ne0 11173  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-map 8822  df-neg 11448  df-nn 12238  df-n0 12509  df-z 12596

This theorem is used by:  eldioph2  43521  eldioph2b  43522