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Theorem prproropf1o 47431
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 47428 . . . 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 9513 . . . . . . 7 (𝑝 = 𝑤 → inf(𝑝, 𝑉, 𝑅) = inf(𝑤, 𝑉, 𝑅))
6 supeq1 9482 . . . . . . 7 (𝑝 = 𝑤 → sup(𝑝, 𝑉, 𝑅) = sup(𝑤, 𝑉, 𝑅))
75, 6opeq12d 4885 . . . . . 6 (𝑝 = 𝑤 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
87cbvmptv 5260 . . . . 5 (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩) = (𝑤𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
94, 8eqtri 2762 . . . 4 𝐹 = (𝑤𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
103, 9fmptd 7133 . . 3 (𝑅 Or 𝑉𝐹:𝑃𝑂)
11 3ancomb 1098 . . . . . 6 ((𝑅 Or 𝑉𝑤𝑃𝑧𝑃) ↔ (𝑅 Or 𝑉𝑧𝑃𝑤𝑃))
12 3anass 1094 . . . . . 6 ((𝑅 Or 𝑉𝑧𝑃𝑤𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧𝑃𝑤𝑃)))
1311, 12bitri 275 . . . . 5 ((𝑅 Or 𝑉𝑤𝑃𝑧𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧𝑃𝑤𝑃)))
141, 2, 4prproropf1olem4 47430 . . . . 5 ((𝑅 Or 𝑉𝑤𝑃𝑧𝑃) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
1513, 14sylbir 235 . . . 4 ((𝑅 Or 𝑉 ∧ (𝑧𝑃𝑤𝑃)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
1615ralrimivva 3199 . . 3 (𝑅 Or 𝑉 → ∀𝑧𝑃𝑤𝑃 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
17 dff13 7274 . . 3 (𝐹:𝑃1-1𝑂 ↔ (𝐹:𝑃𝑂 ∧ ∀𝑧𝑃𝑤𝑃 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
1810, 16, 17sylanbrc 583 . 2 (𝑅 Or 𝑉𝐹:𝑃1-1𝑂)
191, 2prproropf1olem1 47427 . . . . 5 ((𝑅 Or 𝑉𝑤𝑂) → {(1st𝑤), (2nd𝑤)} ∈ 𝑃)
20 fveq2 6906 . . . . . . 7 (𝑧 = {(1st𝑤), (2nd𝑤)} → (𝐹𝑧) = (𝐹‘{(1st𝑤), (2nd𝑤)}))
2120eqeq2d 2745 . . . . . 6 (𝑧 = {(1st𝑤), (2nd𝑤)} → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝐹‘{(1st𝑤), (2nd𝑤)})))
2221adantl 481 . . . . 5 (((𝑅 Or 𝑉𝑤𝑂) ∧ 𝑧 = {(1st𝑤), (2nd𝑤)}) → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝐹‘{(1st𝑤), (2nd𝑤)})))
231, 2, 4prproropf1olem3 47429 . . . . . 6 ((𝑅 Or 𝑉𝑤𝑂) → (𝐹‘{(1st𝑤), (2nd𝑤)}) = ⟨(1st𝑤), (2nd𝑤)⟩)
241prproropf1olem0 47426 . . . . . . . . 9 (𝑤𝑂 ↔ (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ ∧ ((1st𝑤) ∈ 𝑉 ∧ (2nd𝑤) ∈ 𝑉) ∧ (1st𝑤)𝑅(2nd𝑤)))
2524simp1bi 1144 . . . . . . . 8 (𝑤𝑂𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
2625eqcomd 2740 . . . . . . 7 (𝑤𝑂 → ⟨(1st𝑤), (2nd𝑤)⟩ = 𝑤)
2726adantl 481 . . . . . 6 ((𝑅 Or 𝑉𝑤𝑂) → ⟨(1st𝑤), (2nd𝑤)⟩ = 𝑤)
2823, 27eqtr2d 2775 . . . . 5 ((𝑅 Or 𝑉𝑤𝑂) → 𝑤 = (𝐹‘{(1st𝑤), (2nd𝑤)}))
2919, 22, 28rspcedvd 3623 . . . 4 ((𝑅 Or 𝑉𝑤𝑂) → ∃𝑧𝑃 𝑤 = (𝐹𝑧))
3029ralrimiva 3143 . . 3 (𝑅 Or 𝑉 → ∀𝑤𝑂𝑧𝑃 𝑤 = (𝐹𝑧))
31 dffo3 7121 . . 3 (𝐹:𝑃onto𝑂 ↔ (𝐹:𝑃𝑂 ∧ ∀𝑤𝑂𝑧𝑃 𝑤 = (𝐹𝑧)))
3210, 30, 31sylanbrc 583 . 2 (𝑅 Or 𝑉𝐹:𝑃onto𝑂)
33 df-f1o 6569 . 2 (𝐹:𝑃1-1-onto𝑂 ↔ (𝐹:𝑃1-1𝑂𝐹:𝑃onto𝑂))
3418, 32, 33sylanbrc 583 1 (𝑅 Or 𝑉𝐹:𝑃1-1-onto𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  cin 3961  𝒫 cpw 4604  {cpr 4632  cop 4636   class class class wbr 5147  cmpt 5230   Or wor 5595   × cxp 5686  wf 6558  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  1st c1st 8010  2nd c2nd 8011  supcsup 9477  infcinf 9478  2c2 12318  chash 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366
This theorem is referenced by:  prproropen  47432  prproropreud  47433
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