Theorem hashnexinjle 42383

Description: If the number of elements of the domain are greater than the number of elements in a codomain, then there are two different values that map to the same. Also we introduce a one sided inequality to simplify a duplicateable proof. (Contributed by metakunt, 2-May-2025.)

Hypotheses

Ref	Expression
hashnexinjle.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashnexinjle.2	⊢ (𝜑 → 𝐵 ∈ Fin)
hashnexinjle.3	⊢ (𝜑 → (♯‘𝐵) < (♯‘𝐴))
hashnexinjle.4	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
hashnexinjle.5	⊢ (𝜑 → 𝐴 ⊆ ℝ)

Assertion

Ref	Expression
hashnexinjle	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem hashnexinjle

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
2		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
3	2	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
4		breq2 5102	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑧))
5	3, 4	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) ↔ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦 < 𝑧)))
6		fveqeq2 6843	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
7		breq1 5101	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑤 < 𝑧))
8	6, 7	anbi12d 632	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦 < 𝑧) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧)))
9	5, 8	cbvrex2vw 3219	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧))
10	9	a1i 11	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧)))
11	10	biimpd 229	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) → ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧)))
12	11	imp 406	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧))
13		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
14	13	eqeq2d 2747	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
15		breq2 5102	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑤 < 𝑧 ↔ 𝑤 < 𝑦))
16	14, 15	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝑦 → (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧) ↔ ((𝐹‘𝑤) = (𝐹‘𝑦) ∧ 𝑤 < 𝑦)))
17		fveqeq2 6843	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
18		breq1 5101	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 < 𝑦 ↔ 𝑥 < 𝑦))
19	17, 18	anbi12d 632	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝐹‘𝑤) = (𝐹‘𝑦) ∧ 𝑤 < 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)))
20	16, 19	cbvrex2vw 3219	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
21	12, 20	sylib 218	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
22		rexcom 3265	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
23	21, 22	sylib 218	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
24		hashnexinjle.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
25		hashnexinjle.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
26		hashnexinjle.3	. . . 4 ⊢ (𝜑 → (♯‘𝐵) < (♯‘𝐴))
27		hashnexinjle.4	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
28	24, 25, 26, 27	hashnexinj 42382	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))
29		simplrl 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) = (𝐹‘𝑦))
30		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
31	29, 30	jca 511	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
32	31	orcd 873	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
33		simplrl 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (𝐹‘𝑥) = (𝐹‘𝑦))
34	33	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
35		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
36	34, 35	jca 511	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))
37	36	olcd 874	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
38		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
39		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝜑)
40		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
41	39, 40	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝜑 ∧ 𝑥 ∈ 𝐴))
42		hashnexinjle.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
43	42	sselda 3933	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
44	41, 43	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ)
45	44	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ℝ)
46		simprr 772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
47	39, 46	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝜑 ∧ 𝑦 ∈ 𝐴))
48	42	sselda 3933	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
49	47, 48	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ)
50	49	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ℝ)
51	45, 50	lttri2d 11272	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑥 ≠ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)))
52	38, 51	mpbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))
53	32, 37, 52	mpjaodan 960	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
54	53	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))))
55	54	reximdvva 3184	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))))
56	55	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
57		r19.43 3104	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) ↔ (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
58	57	rexbii 3083	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
59	56, 58	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
60		r19.43 3104	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
61	59, 60	sylib 218	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
62	61	ex 412	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))))
63	28, 62	mpd 15	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
64	1, 23, 63	mpjaodan 960	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   ⊆ wss 3901   class class class wbr 5098  ⟶wf 6488  ‘cfv 6492  Fincfn 8883  ℝcr 11025   < clt 11166  ♯chash 14253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254

This theorem is referenced by:  aks6d1c2  42384