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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropen | Structured version Visualization version GIF version |
Description: 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, are equinumerous. (Contributed by AV, 15-Mar-2023.) |
Ref | Expression |
---|---|
prproropf1o.o | ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) |
prproropf1o.p | ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} |
Ref | Expression |
---|---|
prproropen | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → 𝑂 ≈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prproropf1o.p | . . . . . 6 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} | |
2 | pwexg 5331 | . . . . . 6 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V) | |
3 | 1, 2 | rabexd 5288 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → 𝑃 ∈ V) |
5 | 4 | mptexd 7170 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) ∈ V) |
6 | prproropf1o.o | . . . . 5 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) | |
7 | eqid 2737 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) | |
8 | 6, 1, 7 | prproropf1o 45600 | . . . 4 ⊢ (𝑅 Or 𝑉 → (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉):𝑃–1-1-onto→𝑂) |
9 | 8 | adantl 482 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉):𝑃–1-1-onto→𝑂) |
10 | f1oeq1 6769 | . . 3 ⊢ (𝑓 = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) → (𝑓:𝑃–1-1-onto→𝑂 ↔ (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉):𝑃–1-1-onto→𝑂)) | |
11 | 5, 9, 10 | spcedv 3555 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → ∃𝑓 𝑓:𝑃–1-1-onto→𝑂) |
12 | ensymb 8900 | . . 3 ⊢ (𝑂 ≈ 𝑃 ↔ 𝑃 ≈ 𝑂) | |
13 | bren 8851 | . . 3 ⊢ (𝑃 ≈ 𝑂 ↔ ∃𝑓 𝑓:𝑃–1-1-onto→𝑂) | |
14 | 12, 13 | bitri 274 | . 2 ⊢ (𝑂 ≈ 𝑃 ↔ ∃𝑓 𝑓:𝑃–1-1-onto→𝑂) |
15 | 11, 14 | sylibr 233 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → 𝑂 ≈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {crab 3405 Vcvv 3443 ∩ cin 3907 𝒫 cpw 4558 〈cop 4590 class class class wbr 5103 ↦ cmpt 5186 Or wor 5542 × cxp 5629 –1-1-onto→wf1o 6492 ‘cfv 6493 ≈ cen 8838 supcsup 9334 infcinf 9335 2c2 12166 ♯chash 14184 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-oadd 8408 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-hash 14185 |
This theorem is referenced by: (None) |
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