Theorem s3f1 33557

Description: Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)

Hypotheses

Ref	Expression
s3f1.i	⊢ (𝜑 → 𝐼 ∈ 𝐷)
s3f1.j	⊢ (𝜑 → 𝐽 ∈ 𝐷)
s3f1.k	⊢ (𝜑 → 𝐾 ∈ 𝐷)
s3f1.1	⊢ (𝜑 → 𝐼 ≠ 𝐽)
s3f1.2	⊢ (𝜑 → 𝐽 ≠ 𝐾)
s3f1.3	⊢ (𝜑 → 𝐾 ≠ 𝐼)

Assertion

Ref	Expression
s3f1	⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)

Proof of Theorem s3f1

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		s3f1.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
2		s3f1.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
3		s3f1.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
4	1, 2, 3	s3cld 15027	. . . 4 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
5		wrdf 14666	. . . 4 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
7	6	ffdmd 6740	. 2 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩⟶𝐷)
8		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑖 = 0)
9		simpr 490	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑗 = 0)
10	8, 9	eqtr4d 2798	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
11		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
12		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → 𝑖 = 0)
13	12	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘0))
14		s3fv0 15047	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
15	1, 14	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
16	15	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
17	13, 16	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐼)
18	17	adantr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐼)
19		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → 𝑗 = 1)
20	19	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘1))
21		s3fv1 15048	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
22	2, 21	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
23	22	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
24	20, 23	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
25	24	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
26	11, 18, 25	3eqtr3d 2803	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝐼 = 𝐽)
27		s3f1.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
28	27	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝐼 ≠ 𝐽)
29	26, 28	pm2.21ddne 3039	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
30		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
31	17	adantr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐼)
32		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → 𝑗 = 2)
33	32	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘2))
34		s3fv2 15049	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
35	3, 34	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
36	35	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
37	33, 36	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
38	37	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
39	30, 31, 38	3eqtr3rd 2804	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝐾 = 𝐼)
40		s3f1.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≠ 𝐼)
41	40	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝐾 ≠ 𝐼)
42	39, 41	pm2.21ddne 3039	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
43		wrddm 14669	. . . . . . . . . . . . . 14 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
44	4, 43	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
45		s3len 15050	. . . . . . . . . . . . . . 15 ⊢ (♯‘⟨“𝐼𝐽𝐾”⟩) = 3
46	45	oveq2i 7431	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = (0..^3)
47		fzo0to3tp 13890	. . . . . . . . . . . . . 14 ⊢ (0..^3) = {0, 1, 2}
48	46, 47	eqtri 2783	. . . . . . . . . . . . 13 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = {0, 1, 2}
49	44, 48	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = {0, 1, 2})
50	49	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑗 ∈ {0, 1, 2}))
51	50	biimpa 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑗 ∈ {0, 1, 2})
52		vex 3454	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
53	52	eltp 4646	. . . . . . . . . 10 ⊢ (𝑗 ∈ {0, 1, 2} ↔ (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
54	51, 53	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
55	54	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
56	55	ad2antrr 739	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
57	10, 29, 42, 56	mpjao3dan 1459	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → 𝑖 = 𝑗)
58		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
59		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → 𝑖 = 1)
60	59	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘1))
61	22	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
62	60, 61	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐽)
63	62	adantr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐽)
64		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → 𝑗 = 0)
65	64	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘0))
66	15	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
67	65, 66	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
68	67	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
69	58, 63, 68	3eqtr3rd 2804	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝐼 = 𝐽)
70	27	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝐼 ≠ 𝐽)
71	69, 70	pm2.21ddne 3039	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
72		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑖 = 1)
73		simpr 490	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑗 = 1)
74	72, 73	eqtr4d 2798	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
75		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
76	62	adantr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐽)
77	37	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
78	75, 76, 77	3eqtr3d 2803	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝐽 = 𝐾)
79		s3f1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ≠ 𝐾)
80	79	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝐽 ≠ 𝐾)
81	78, 80	pm2.21ddne 3039	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
82	55	ad2antrr 739	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
83	71, 74, 81, 82	mpjao3dan 1459	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → 𝑖 = 𝑗)
84		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
85		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → 𝑖 = 2)
86	85	fveq2d 6889	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘2))
87	35	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
88	86, 87	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐾)
89	88	adantr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐾)
90	67	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
91	84, 89, 90	3eqtr3d 2803	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝐾 = 𝐼)
92	40	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝐾 ≠ 𝐼)
93	91, 92	pm2.21ddne 3039	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
94		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
95	88	adantr 486	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐾)
96	24	adantlr 728	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
97	94, 95, 96	3eqtr3rd 2804	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝐽 = 𝐾)
98	79	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝐽 ≠ 𝐾)
99	97, 98	pm2.21ddne 3039	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
100		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑖 = 2)
101		simpr 490	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑗 = 2)
102	100, 101	eqtr4d 2798	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
103	55	ad2antrr 739	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
104	93, 99, 102, 103	mpjao3dan 1459	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → 𝑖 = 𝑗)
105	49	eleq2d 2846	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑖 ∈ {0, 1, 2}))
106	105	biimpa 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑖 ∈ {0, 1, 2})
107		vex 3454	. . . . . . . . 9 ⊢ 𝑖 ∈ V
108	107	eltp 4646	. . . . . . . 8 ⊢ (𝑖 ∈ {0, 1, 2} ↔ (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
109	106, 108	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
110	109	ad2antrr 739	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
111	57, 83, 104, 110	mpjao3dan 1459	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) → 𝑖 = 𝑗)
112	111	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → ((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
113	112	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩)) → ((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
114	113	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩∀𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
115		dff13 7258	. 2 ⊢ (⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷 ↔ (⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩⟶𝐷 ∧ ∀𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩∀𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗)))
116	7, 114, 115	sylanbrc 595	1 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {ctp 4588  dom cdm 5650  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  0cc0 11203  1c1 11204  2c2 12400  3c3 12401  ..^cfzo 13791  ♯chash 14477  Word cword 14661  ⟨“cs3 14997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-z 12697  df-uz 12969  df-fz 13643  df-fzo 13792  df-hash 14478  df-word 14662  df-concat 14719  df-s1 14746  df-s2 15003  df-s3 15004

This theorem is used by:  cycpm3cl  33742  cycpm3cl2  33743  cyc3fv1  33744  cyc3fv2  33745  cyc3fv3  33746  cyc3co2  33747