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Theorem hashnexinjle 42130
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 (𝜑𝐴 ∈ Fin)
hashnexinjle.2 (𝜑𝐵 ∈ Fin)
hashnexinjle.3 (𝜑 → (♯‘𝐵) < (♯‘𝐴))
hashnexinjle.4 (𝜑𝐹:𝐴𝐵)
hashnexinjle.5 (𝜑𝐴 ⊆ ℝ)
Assertion
Ref Expression
hashnexinjle (𝜑 → ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 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.
StepHypRef Expression
1 simpr 484 . 2 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
2 fveq2 6906 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
32eqeq2d 2748 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝐹𝑦) = (𝐹𝑥) ↔ (𝐹𝑦) = (𝐹𝑧)))
4 breq2 5147 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑦 < 𝑥𝑦 < 𝑧))
53, 4anbi12d 632 . . . . . . . 8 (𝑥 = 𝑧 → (((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥) ↔ ((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦 < 𝑧)))
6 fveqeq2 6915 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝐹𝑦) = (𝐹𝑧) ↔ (𝐹𝑤) = (𝐹𝑧)))
7 breq1 5146 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦 < 𝑧𝑤 < 𝑧))
86, 7anbi12d 632 . . . . . . . 8 (𝑦 = 𝑤 → (((𝐹𝑦) = (𝐹𝑧) ∧ 𝑦 < 𝑧) ↔ ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧)))
95, 8cbvrex2vw 3242 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥) ↔ ∃𝑧𝐴𝑤𝐴 ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧))
109a1i 11 . . . . . 6 (𝜑 → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥) ↔ ∃𝑧𝐴𝑤𝐴 ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧)))
1110biimpd 229 . . . . 5 (𝜑 → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥) → ∃𝑧𝐴𝑤𝐴 ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧)))
1211imp 406 . . . 4 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑧𝐴𝑤𝐴 ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧))
13 fveq2 6906 . . . . . . 7 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
1413eqeq2d 2748 . . . . . 6 (𝑧 = 𝑦 → ((𝐹𝑤) = (𝐹𝑧) ↔ (𝐹𝑤) = (𝐹𝑦)))
15 breq2 5147 . . . . . 6 (𝑧 = 𝑦 → (𝑤 < 𝑧𝑤 < 𝑦))
1614, 15anbi12d 632 . . . . 5 (𝑧 = 𝑦 → (((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧) ↔ ((𝐹𝑤) = (𝐹𝑦) ∧ 𝑤 < 𝑦)))
17 fveqeq2 6915 . . . . . 6 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
18 breq1 5146 . . . . . 6 (𝑤 = 𝑥 → (𝑤 < 𝑦𝑥 < 𝑦))
1917, 18anbi12d 632 . . . . 5 (𝑤 = 𝑥 → (((𝐹𝑤) = (𝐹𝑦) ∧ 𝑤 < 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)))
2016, 19cbvrex2vw 3242 . . . 4 (∃𝑧𝐴𝑤𝐴 ((𝐹𝑤) = (𝐹𝑧) ∧ 𝑤 < 𝑧) ↔ ∃𝑦𝐴𝑥𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
2112, 20sylib 218 . . 3 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑦𝐴𝑥𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
22 rexcom 3290 . . 3 (∃𝑦𝐴𝑥𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ↔ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
2321, 22sylib 218 . 2 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
24 hashnexinjle.1 . . . 4 (𝜑𝐴 ∈ Fin)
25 hashnexinjle.2 . . . 4 (𝜑𝐵 ∈ Fin)
26 hashnexinjle.3 . . . 4 (𝜑 → (♯‘𝐵) < (♯‘𝐴))
27 hashnexinjle.4 . . . 4 (𝜑𝐹:𝐴𝐵)
2824, 25, 26, 27hashnexinj 42129 . . 3 (𝜑 → ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦))
29 simplrl 777 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑥 < 𝑦) → (𝐹𝑥) = (𝐹𝑦))
30 simpr 484 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
3129, 30jca 511 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑥 < 𝑦) → ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
3231orcd 874 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑥 < 𝑦) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
33 simplrl 777 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑦 < 𝑥) → (𝐹𝑥) = (𝐹𝑦))
3433eqcomd 2743 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑦 < 𝑥) → (𝐹𝑦) = (𝐹𝑥))
35 simpr 484 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
3634, 35jca 511 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑦 < 𝑥) → ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥))
3736olcd 875 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) ∧ 𝑦 < 𝑥) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
38 simprr 773 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
39 simpl 482 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝜑)
40 simprl 771 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
4139, 40jca 511 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝜑𝑥𝐴))
42 hashnexinjle.5 . . . . . . . . . . . . . . 15 (𝜑𝐴 ⊆ ℝ)
4342sselda 3983 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑥 ∈ ℝ)
4441, 43syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 ∈ ℝ)
4544adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ ℝ)
46 simprr 773 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
4739, 46jca 511 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝜑𝑦𝐴))
4842sselda 3983 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → 𝑦 ∈ ℝ)
4947, 48syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑦 ∈ ℝ)
5049adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → 𝑦 ∈ ℝ)
5145, 50lttri2d 11400 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → (𝑥𝑦 ↔ (𝑥 < 𝑦𝑦 < 𝑥)))
5238, 51mpbid 232 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → (𝑥 < 𝑦𝑦 < 𝑥))
5332, 37, 52mpjaodan 961 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
5453ex 412 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥))))
5554reximdvva 3207 . . . . . . 7 (𝜑 → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥))))
5655imp 406 . . . . . 6 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → ∃𝑥𝐴𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
57 r19.43 3122 . . . . . . 7 (∃𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)) ↔ (∃𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
5857rexbii 3094 . . . . . 6 (∃𝑥𝐴𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)) ↔ ∃𝑥𝐴 (∃𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
5956, 58sylib 218 . . . . 5 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → ∃𝑥𝐴 (∃𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
60 r19.43 3122 . . . . 5 (∃𝑥𝐴 (∃𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)) ↔ (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
6159, 60sylib 218 . . . 4 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)) → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
6261ex 412 . . 3 (𝜑 → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦) → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥))))
6328, 62mpd 15 . 2 (𝜑 → (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑦) = (𝐹𝑥) ∧ 𝑦 < 𝑥)))
641, 23, 63mpjaodan 961 1 (𝜑 → ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wrex 3070  wss 3951   class class class wbr 5143  wf 6557  cfv 6561  Fincfn 8985  cr 11154   < clt 11295  chash 14369
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-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-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-oadd 8510  df-er 8745  df-map 8868  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-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370
This theorem is referenced by:  aks6d1c2  42131
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