Theorem hashnexinjle 42105

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 6826	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
3	2	eqeq2d 2740	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
4		breq2 5099	. . . . . . . . 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 6835	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
7		breq1 5098	. . . . . . . . 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 3212	. . . . . . 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 6826	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
14	13	eqeq2d 2740	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
15		breq2 5099	. . . . . 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 6835	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
18		breq1 5098	. . . . . 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 3212	. . . 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 3258	. . 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 42104	. . 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 2735	. . . . . . . . . . . 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 3937	. . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . 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 11273	. . . . . . . . . . 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 3177	. . . . . . 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 3097	. . . . . . 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 3076	. . . . . 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 3097	. . . . 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 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095  ⟶wf 6482  ‘cfv 6486  Fincfn 8879  ℝcr 11027   < clt 11168  ♯chash 14255

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12754  df-fz 13429  df-hash 14256

This theorem is referenced by:  aks6d1c2  42106