Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prproropf1o Structured version   Visualization version   GIF version

Theorem prproropf1o 44039
 Description: There is a bijection between the set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component. (Contributed by AV, 15-Mar-2023.)
Hypotheses
Ref Expression
prproropf1o.o 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
prproropf1o.p 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
prproropf1o.f 𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
Assertion
Ref Expression
prproropf1o (𝑅 Or 𝑉𝐹:𝑃1-1-onto𝑂)
Distinct variable groups:   𝑉,𝑝   𝑂,𝑝   𝑃,𝑝   𝑅,𝑝
Allowed substitution hint:   𝐹(𝑝)

Proof of Theorem prproropf1o
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prproropf1o.o . . . . 5 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
2 prproropf1o.p . . . . 5 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
31, 2prproropf1olem2 44036 . . . 4 ((𝑅 Or 𝑉𝑤𝑃) → ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩ ∈ 𝑂)
4 prproropf1o.f . . . . 5 𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
5 infeq1 8926 . . . . . . 7 (𝑝 = 𝑤 → inf(𝑝, 𝑉, 𝑅) = inf(𝑤, 𝑉, 𝑅))
6 supeq1 8895 . . . . . . 7 (𝑝 = 𝑤 → sup(𝑝, 𝑉, 𝑅) = sup(𝑤, 𝑉, 𝑅))
75, 6opeq12d 4773 . . . . . 6 (𝑝 = 𝑤 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
87cbvmptv 5133 . . . . 5 (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩) = (𝑤𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
94, 8eqtri 2821 . . . 4 𝐹 = (𝑤𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
103, 9fmptd 6855 . . 3 (𝑅 Or 𝑉𝐹:𝑃𝑂)
11 3ancomb 1096 . . . . . 6 ((𝑅 Or 𝑉𝑤𝑃𝑧𝑃) ↔ (𝑅 Or 𝑉𝑧𝑃𝑤𝑃))
12 3anass 1092 . . . . . 6 ((𝑅 Or 𝑉𝑧𝑃𝑤𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧𝑃𝑤𝑃)))
1311, 12bitri 278 . . . . 5 ((𝑅 Or 𝑉𝑤𝑃𝑧𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧𝑃𝑤𝑃)))
141, 2, 4prproropf1olem4 44038 . . . . 5 ((𝑅 Or 𝑉𝑤𝑃𝑧𝑃) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
1513, 14sylbir 238 . . . 4 ((𝑅 Or 𝑉 ∧ (𝑧𝑃𝑤𝑃)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
1615ralrimivva 3156 . . 3 (𝑅 Or 𝑉 → ∀𝑧𝑃𝑤𝑃 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
17 dff13 6991 . . 3 (𝐹:𝑃1-1𝑂 ↔ (𝐹:𝑃𝑂 ∧ ∀𝑧𝑃𝑤𝑃 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
1810, 16, 17sylanbrc 586 . 2 (𝑅 Or 𝑉𝐹:𝑃1-1𝑂)
191, 2prproropf1olem1 44035 . . . . 5 ((𝑅 Or 𝑉𝑤𝑂) → {(1st𝑤), (2nd𝑤)} ∈ 𝑃)
20 fveq2 6645 . . . . . . 7 (𝑧 = {(1st𝑤), (2nd𝑤)} → (𝐹𝑧) = (𝐹‘{(1st𝑤), (2nd𝑤)}))
2120eqeq2d 2809 . . . . . 6 (𝑧 = {(1st𝑤), (2nd𝑤)} → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝐹‘{(1st𝑤), (2nd𝑤)})))
2221adantl 485 . . . . 5 (((𝑅 Or 𝑉𝑤𝑂) ∧ 𝑧 = {(1st𝑤), (2nd𝑤)}) → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝐹‘{(1st𝑤), (2nd𝑤)})))
231, 2, 4prproropf1olem3 44037 . . . . . 6 ((𝑅 Or 𝑉𝑤𝑂) → (𝐹‘{(1st𝑤), (2nd𝑤)}) = ⟨(1st𝑤), (2nd𝑤)⟩)
241prproropf1olem0 44034 . . . . . . . . 9 (𝑤𝑂 ↔ (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ ∧ ((1st𝑤) ∈ 𝑉 ∧ (2nd𝑤) ∈ 𝑉) ∧ (1st𝑤)𝑅(2nd𝑤)))
2524simp1bi 1142 . . . . . . . 8 (𝑤𝑂𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
2625eqcomd 2804 . . . . . . 7 (𝑤𝑂 → ⟨(1st𝑤), (2nd𝑤)⟩ = 𝑤)
2726adantl 485 . . . . . 6 ((𝑅 Or 𝑉𝑤𝑂) → ⟨(1st𝑤), (2nd𝑤)⟩ = 𝑤)
2823, 27eqtr2d 2834 . . . . 5 ((𝑅 Or 𝑉𝑤𝑂) → 𝑤 = (𝐹‘{(1st𝑤), (2nd𝑤)}))
2919, 22, 28rspcedvd 3574 . . . 4 ((𝑅 Or 𝑉𝑤𝑂) → ∃𝑧𝑃 𝑤 = (𝐹𝑧))
3029ralrimiva 3149 . . 3 (𝑅 Or 𝑉 → ∀𝑤𝑂𝑧𝑃 𝑤 = (𝐹𝑧))
31 dffo3 6845 . . 3 (𝐹:𝑃onto𝑂 ↔ (𝐹:𝑃𝑂 ∧ ∀𝑤𝑂𝑧𝑃 𝑤 = (𝐹𝑧)))
3210, 30, 31sylanbrc 586 . 2 (𝑅 Or 𝑉𝐹:𝑃onto𝑂)
33 df-f1o 6331 . 2 (𝐹:𝑃1-1-onto𝑂 ↔ (𝐹:𝑃1-1𝑂𝐹:𝑃onto𝑂))
3418, 32, 33sylanbrc 586 1 (𝑅 Or 𝑉𝐹:𝑃1-1-onto𝑂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ∩ cin 3880  𝒫 cpw 4497  {cpr 4527  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   Or wor 5437   × cxp 5517  ⟶wf 6320  –1-1→wf1 6321  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324  1st c1st 7671  2nd c2nd 7672  supcsup 8890  infcinf 8891  2c2 11682  ♯chash 13688 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-sup 8892  df-inf 8893  df-dju 9316  df-card 9354  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-nn 11628  df-2 11690  df-n0 11888  df-z 11972  df-uz 12234  df-fz 12888  df-hash 13689 This theorem is referenced by:  prproropen  44040  prproropreud  44041
 Copyright terms: Public domain W3C validator