Theorem hashnexinjle 42496

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 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
3	2	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
4		breq2 5104	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑧))
5	3, 4	anbi12d 633	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) ↔ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦 < 𝑧)))
6		fveqeq2 6851	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
7		breq1 5103	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑤 < 𝑧))
8	6, 7	anbi12d 633	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦 < 𝑧) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧)))
9	5, 8	cbvrex2vw 3221	. . . . . . 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 6842	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
14	13	eqeq2d 2748	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
15		breq2 5104	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑤 < 𝑧 ↔ 𝑤 < 𝑦))
16	14, 15	anbi12d 633	. . . . 5 ⊢ (𝑧 = 𝑦 → (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧) ↔ ((𝐹‘𝑤) = (𝐹‘𝑦) ∧ 𝑤 < 𝑦)))
17		fveqeq2 6851	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
18		breq1 5103	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 < 𝑦 ↔ 𝑥 < 𝑦))
19	17, 18	anbi12d 633	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝐹‘𝑤) = (𝐹‘𝑦) ∧ 𝑤 < 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)))
20	16, 19	cbvrex2vw 3221	. . . 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 3267	. . 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 42495	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))
29		simplrl 777	. . . . . . . . . . . 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 874	. . . . . . . . . 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 777	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (𝐹‘𝑥) = (𝐹‘𝑦))
34	33	eqcomd 2743	. . . . . . . . . . . 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 875	. . . . . . . . . 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 773	. . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . 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 11284	. . . . . . . . . . 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 961	. . . . . . . . 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 3186	. . . . . . 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 3106	. . . . . . 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 3085	. . . . . 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 3106	. . . . 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 961	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100  ⟶wf 6496  ‘cfv 6500  Fincfn 8895  ℝcr 11037   < clt 11178  ♯chash 14265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-fz 13436  df-hash 14266

This theorem is referenced by:  aks6d1c2  42497