Theorem s3f1 31123

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 14513	. . . 4 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
5		wrdf 14150	. . . 4 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
7	6	ffdmd 6615	. 2 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩⟶𝐷)
8		simplr 765	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑖 = 0)
9		simpr 484	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑗 = 0)
10	8, 9	eqtr4d 2781	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
11		simpllr 772	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
12		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → 𝑖 = 0)
13	12	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘0))
14		s3fv0 14532	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
15	1, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
16	15	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
17	13, 16	eqtrd 2778	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐼)
18	17	adantr 480	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐼)
19		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → 𝑗 = 1)
20	19	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘1))
21		s3fv1 14533	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
22	2, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
23	22	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
24	20, 23	eqtrd 2778	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
25	24	adantlr 711	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
26	11, 18, 25	3eqtr3d 2786	. . . . . . . 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 730	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝐼 ≠ 𝐽)
29	26, 28	pm2.21ddne 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
30		simpllr 772	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
31	17	adantr 480	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐼)
32		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → 𝑗 = 2)
33	32	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘2))
34		s3fv2 14534	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
35	3, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
36	35	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
37	33, 36	eqtrd 2778	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
38	37	adantlr 711	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
39	30, 31, 38	3eqtr3rd 2787	. . . . . . . 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 730	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝐾 ≠ 𝐼)
42	39, 41	pm2.21ddne 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
43		wrddm 14152	. . . . . . . . . . . . . 14 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
44	4, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
45		s3len 14535	. . . . . . . . . . . . . . 15 ⊢ (♯‘⟨“𝐼𝐽𝐾”⟩) = 3
46	45	oveq2i 7266	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = (0..^3)
47		fzo0to3tp 13401	. . . . . . . . . . . . . 14 ⊢ (0..^3) = {0, 1, 2}
48	46, 47	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = {0, 1, 2}
49	44, 48	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = {0, 1, 2})
50	49	eleq2d 2824	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑗 ∈ {0, 1, 2}))
51	50	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑗 ∈ {0, 1, 2})
52		vex 3426	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
53	52	eltp 4621	. . . . . . . . . 10 ⊢ (𝑗 ∈ {0, 1, 2} ↔ (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
54	51, 53	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
55	54	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
56	55	ad2antrr 722	. . . . . . 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 1429	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → 𝑖 = 𝑗)
58		simpllr 772	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
59		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → 𝑖 = 1)
60	59	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘1))
61	22	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
62	60, 61	eqtrd 2778	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐽)
63	62	adantr 480	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐽)
64		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → 𝑗 = 0)
65	64	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘0))
66	15	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
67	65, 66	eqtrd 2778	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
68	67	adantlr 711	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
69	58, 63, 68	3eqtr3rd 2787	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝐼 = 𝐽)
70	27	ad5antr 730	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝐼 ≠ 𝐽)
71	69, 70	pm2.21ddne 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
72		simplr 765	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑖 = 1)
73		simpr 484	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑗 = 1)
74	72, 73	eqtr4d 2781	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
75		simpllr 772	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
76	62	adantr 480	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐽)
77	37	adantlr 711	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
78	75, 76, 77	3eqtr3d 2786	. . . . . . . 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 730	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝐽 ≠ 𝐾)
81	78, 80	pm2.21ddne 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
82	55	ad2antrr 722	. . . . . . 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 1429	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → 𝑖 = 𝑗)
84		simpllr 772	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
85		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → 𝑖 = 2)
86	85	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘2))
87	35	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
88	86, 87	eqtrd 2778	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐾)
89	88	adantr 480	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐾)
90	67	adantlr 711	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
91	84, 89, 90	3eqtr3d 2786	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝐾 = 𝐼)
92	40	ad5antr 730	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝐾 ≠ 𝐼)
93	91, 92	pm2.21ddne 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
94		simpllr 772	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗))
95	88	adantr 480	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = 𝐾)
96	24	adantlr 711	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
97	94, 95, 96	3eqtr3rd 2787	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝐽 = 𝐾)
98	79	ad5antr 730	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝐽 ≠ 𝐾)
99	97, 98	pm2.21ddne 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
100		simplr 765	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑖 = 2)
101		simpr 484	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑗 = 2)
102	100, 101	eqtr4d 2781	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
103	55	ad2antrr 722	. . . . . . 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 1429	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → 𝑖 = 𝑗)
105	49	eleq2d 2824	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑖 ∈ {0, 1, 2}))
106	105	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑖 ∈ {0, 1, 2})
107		vex 3426	. . . . . . . . 9 ⊢ 𝑖 ∈ V
108	107	eltp 4621	. . . . . . . 8 ⊢ (𝑖 ∈ {0, 1, 2} ↔ (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
109	106, 108	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
110	109	ad2antrr 722	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
111	57, 83, 104, 110	mpjao3dan 1429	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) → 𝑖 = 𝑗)
112	111	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → ((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
113	112	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩)) → ((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
114	113	ralrimivva 3114	. 2 ⊢ (𝜑 → ∀𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩∀𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
115		dff13 7109	. 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 582	1 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1084   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {ctp 4562  dom cdm 5580  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803  2c2 11958  3c3 11959  ..^cfzo 13311  ♯chash 13972  Word cword 14145  ⟨“cs3 14483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490

This theorem is referenced by:  cycpm3cl  31304  cycpm3cl2  31305  cyc3fv1  31306  cyc3fv2  31307  cyc3fv3  31308  cyc3co2  31309