Theorem s3f1 32916

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 14908	. . . 4 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
5		wrdf 14554	. . . 4 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:(0..^(♯‘⟨“𝐼𝐽𝐾”⟩))⟶𝐷)
7	6	ffdmd 6767	. 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 2778	. . . . . . 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 6911	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) → (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘0))
14		s3fv0 14927	. . . . . . . . . . . . 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 2775	. . . . . . . . . 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 6911	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 1) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘1))
21		s3fv1 14928	. . . . . . . . . . . . 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 2775	. . . . . . . . . 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 2783	. . . . . . . 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 3024	. . . . . . 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 6911	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑗 = 2) → (⟨“𝐼𝐽𝐾”⟩‘𝑗) = (⟨“𝐼𝐽𝐾”⟩‘2))
34		s3fv2 14929	. . . . . . . . . . . . 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 2775	. . . . . . . . . 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 2784	. . . . . . . 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 3024	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 0) ∧ 𝑗 = 2) → 𝑖 = 𝑗)
43		wrddm 14556	. . . . . . . . . . . . . 14 ⊢ (⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
44	4, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)))
45		s3len 14930	. . . . . . . . . . . . . . 15 ⊢ (♯‘⟨“𝐼𝐽𝐾”⟩) = 3
46	45	oveq2i 7442	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = (0..^3)
47		fzo0to3tp 13788	. . . . . . . . . . . . . 14 ⊢ (0..^3) = {0, 1, 2}
48	46, 47	eqtri 2763	. . . . . . . . . . . . 13 ⊢ (0..^(♯‘⟨“𝐼𝐽𝐾”⟩)) = {0, 1, 2}
49	44, 48	eqtrdi 2791	. . . . . . . . . . . 12 ⊢ (𝜑 → dom ⟨“𝐼𝐽𝐾”⟩ = {0, 1, 2})
50	49	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑗 ∈ {0, 1, 2}))
51	50	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑗 ∈ {0, 1, 2})
52		vex 3482	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
53	52	eltp 4694	. . . . . . . . . 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 1431	. . . . . 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 6911	. . . . . . . . . . 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 2775	. . . . . . . . . 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 6911	. . . . . . . . . . 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 2775	. . . . . . . . . 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 2784	. . . . . . . 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 3024	. . . . . . 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 2778	. . . . . . 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 2783	. . . . . . . 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 3024	. . . . . . 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 1431	. . . . . 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 6911	. . . . . . . . . . 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 2775	. . . . . . . . . 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 2783	. . . . . . . 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 3024	. . . . . . 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 2784	. . . . . . . 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 3024	. . . . . . 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 2778	. . . . . . 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 1431	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ 𝑗 ∈ dom ⟨“𝐼𝐽𝐾”⟩) ∧ (⟨“𝐼𝐽𝐾”⟩‘𝑖) = (⟨“𝐼𝐽𝐾”⟩‘𝑗)) ∧ 𝑖 = 2) → 𝑖 = 𝑗)
105	49	eleq2d 2825	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩ ↔ 𝑖 ∈ {0, 1, 2}))
106	105	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom ⟨“𝐼𝐽𝐾”⟩) → 𝑖 ∈ {0, 1, 2})
107		vex 3482	. . . . . . . . 9 ⊢ 𝑖 ∈ V
108	107	eltp 4694	. . . . . . . 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 1431	. . . . 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 3200	. 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)

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {ctp 4635  dom cdm 5689  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  2c2 12319  3c3 12320  ..^cfzo 13691  ♯chash 14366  Word cword 14549  ⟨“cs3 14878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885

This theorem is referenced by:  cycpm3cl  33138  cycpm3cl2  33139  cyc3fv1  33140  cyc3fv2  33141  cyc3fv3  33142  cyc3co2  33143