s3f1 - Mathbox for Thierry Arnoux

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Theorem s3f1 32931

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 14911	. . . 4 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
5		wrdf 14557	. . . 4 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
7	6	ffdmd 6766	. 2 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩⟶𝐷)
8		simplr 769	. . . . . . . 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 2780	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
11		simpllr 776	. . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘0))
14		s3fv0 14930	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
15	1, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
16	15	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
17	13, 16	eqtrd 2777	. . . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘1))
21		s3fv1 14931	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
22	2, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
23	22	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
24	20, 23	eqtrd 2777	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
25	24	adantlr 715	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
26	11, 18, 25	3eqtr3d 2785	. . . . . . . 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 734	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝐼 ≠ 𝐽)
29	26, 28	pm2.21ddne 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
30		simpllr 776	. . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘2))
34		s3fv2 14932	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝐷 → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
35	3, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
36	35	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
37	33, 36	eqtrd 2777	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
38	37	adantlr 715	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
39	30, 31, 38	3eqtr3rd 2786	. . . . . . . 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 734	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝐾 ≠ 𝐼)
42	39, 41	pm2.21ddne 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
43		wrddm 14559	. . . . . . . . . . . . . 14 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
44	4, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
45		s3len 14933	. . . . . . . . . . . . . . 15 ⊢ (♯‘⟨“𝐼𝐽𝐾”⟩) = 3
46	45	oveq2i 7442	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = (0..^3)
47		fzo0to3tp 13791	. . . . . . . . . . . . . 14 ⊢ (0..^3) = {0, 1, 2}
48	46, 47	eqtri 2765	. . . . . . . . . . . . 13 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = {0, 1, 2}
49	44, 48	eqtrdi 2793	. . . . . . . . . . . 12 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = {0, 1, 2})
50	49	eleq2d 2827	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑗 ∈ {0, 1, 2}))
51	50	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑗 ∈ {0, 1, 2})
52		vex 3484	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
53	52	eltp 4689	. . . . . . . . . 10 ⊢ (𝑗 ∈ {0, 1, 2} ↔ (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
54	51, 53	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
55	54	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑗 = 0 ∨ 𝑗 = 1 ∨ 𝑗 = 2))
56	55	ad2antrr 726	. . . . . . 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 1434	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → 𝑖 = 𝑗)
58		simpllr 776	. . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘1))
61	22	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → (⟨“𝐼𝐽𝐾”⟩‘1) = 𝐽)
62	60, 61	eqtrd 2777	. . . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘0))
66	15	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘0) = 𝐼)
67	65, 66	eqtrd 2777	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
68	67	adantlr 715	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
69	58, 63, 68	3eqtr3rd 2786	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝐼 = 𝐽)
70	27	ad5antr 734	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝐼 ≠ 𝐽)
71	69, 70	pm2.21ddne 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
72		simplr 769	. . . . . . . 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 2780	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
75		simpllr 776	. . . . . . . . 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 715	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐾)
78	75, 76, 77	3eqtr3d 2785	. . . . . . . 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 734	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝐽 ≠ 𝐾)
81	78, 80	pm2.21ddne 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
82	55	ad2antrr 726	. . . . . . 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 1434	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 1) → 𝑖 = 𝑗)
84		simpllr 776	. . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘2))
87	35	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → (⟨“𝐼𝐽𝐾”⟩‘2) = 𝐾)
88	86, 87	eqtrd 2777	. . . . . . . . . 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 715	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐼)
91	84, 89, 90	3eqtr3d 2785	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝐾 = 𝐼)
92	40	ad5antr 734	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝐾 ≠ 𝐼)
93	91, 92	pm2.21ddne 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 0) → 𝑖 = 𝑗)
94		simpllr 776	. . . . . . . . 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 715	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = 𝐽)
97	94, 95, 96	3eqtr3rd 2786	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝐽 = 𝐾)
98	79	ad5antr 734	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝐽 ≠ 𝐾)
99	97, 98	pm2.21ddne 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 1) → 𝑖 = 𝑗)
100		simplr 769	. . . . . . . 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 2780	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
103	55	ad2antrr 726	. . . . . . 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 1434	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → 𝑖 = 𝑗)
105	49	eleq2d 2827	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑖 ∈ {0, 1, 2}))
106	105	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑖 ∈ {0, 1, 2})
107		vex 3484	. . . . . . . . 9 ⊢ 𝑖 ∈ V
108	107	eltp 4689	. . . . . . . 8 ⊢ (𝑖 ∈ {0, 1, 2} ↔ (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
109	106, 108	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
110	109	ad2antrr 726	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
111	57, 83, 104, 110	mpjao3dan 1434	. . . . 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 3202	. 2 ⊢ (𝜑 → ∀𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩∀𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩((⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗) → 𝑖 = 𝑗))
115		dff13 7275	. 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 583	1 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {ctp 4630 dom cdm 5685 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 2c2 12321 3c3 12322 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ⟨“cs3 14881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888

This theorem is referenced by: cycpm3cl 33155 cycpm3cl2 33156 cyc3fv1 33157 cyc3fv2 33158 cyc3fv3 33159 cyc3co2 33160

W3C validator