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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1o | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
prproropf1o.o | ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) |
prproropf1o.p | ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} |
prproropf1o.f | ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) |
Ref | Expression |
---|---|
prproropf1o | ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prproropf1o.o | . . . . 5 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) | |
2 | prproropf1o.p | . . . . 5 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} | |
3 | 1, 2 | prproropf1olem2 47428 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃) → 〈inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)〉 ∈ 𝑂) |
4 | prproropf1o.f | . . . . 5 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ 〈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)) | |
7 | 5, 6 | opeq12d 4885 | . . . . . 6 ⊢ (𝑝 = 𝑤 → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)〉) |
8 | 7 | cbvmptv 5260 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) = (𝑤 ∈ 𝑃 ↦ 〈inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)〉) |
9 | 4, 8 | eqtri 2762 | . . . 4 ⊢ 𝐹 = (𝑤 ∈ 𝑃 ↦ 〈inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)〉) |
10 | 3, 9 | fmptd 7133 | . . 3 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃⟶𝑂) |
11 | 3ancomb 1098 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)) | |
12 | 3anass 1094 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))) | |
13 | 11, 12 | bitri 275 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))) |
14 | 1, 2, 4 | prproropf1olem4 47430 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
15 | 13, 14 | sylbir 235 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
16 | 15 | ralrimivva 3199 | . . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
17 | dff13 7274 | . . 3 ⊢ (𝐹:𝑃–1-1→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))) | |
18 | 10, 16, 17 | sylanbrc 583 | . 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1→𝑂) |
19 | 1, 2 | prproropf1olem1 47427 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → {(1st ‘𝑤), (2nd ‘𝑤)} ∈ 𝑃) |
20 | fveq2 6906 | . . . . . . 7 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝐹‘𝑧) = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})) | |
21 | 20 | eqeq2d 2745 | . . . . . 6 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))) |
22 | 21 | adantl 481 | . . . . 5 ⊢ (((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) ∧ 𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)}) → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))) |
23 | 1, 2, 4 | prproropf1olem3 47429 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}) = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
24 | 1 | prproropf1olem0 47426 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝑂 ↔ (𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∧ ((1st ‘𝑤) ∈ 𝑉 ∧ (2nd ‘𝑤) ∈ 𝑉) ∧ (1st ‘𝑤)𝑅(2nd ‘𝑤))) |
25 | 24 | simp1bi 1144 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝑂 → 𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
26 | 25 | eqcomd 2740 | . . . . . . 7 ⊢ (𝑤 ∈ 𝑂 → 〈(1st ‘𝑤), (2nd ‘𝑤)〉 = 𝑤) |
27 | 26 | adantl 481 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 〈(1st ‘𝑤), (2nd ‘𝑤)〉 = 𝑤) |
28 | 23, 27 | eqtr2d 2775 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})) |
29 | 19, 22, 28 | rspcedvd 3623 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧)) |
30 | 29 | ralrimiva 3143 | . . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧)) |
31 | dffo3 7121 | . . 3 ⊢ (𝐹:𝑃–onto→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))) | |
32 | 10, 30, 31 | sylanbrc 583 | . 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–onto→𝑂) |
33 | df-f1o 6569 | . 2 ⊢ (𝐹:𝑃–1-1-onto→𝑂 ↔ (𝐹:𝑃–1-1→𝑂 ∧ 𝐹:𝑃–onto→𝑂)) | |
34 | 18, 32, 33 | sylanbrc 583 | 1 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂) |
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 –1-1→wf1 6559 –onto→wfo 6560 –1-1-onto→wf1o 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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