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 43673 | . . . 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 8943 | . . . . . . 7 ⊢ (𝑝 = 𝑤 → inf(𝑝, 𝑉, 𝑅) = inf(𝑤, 𝑉, 𝑅)) | |
6 | supeq1 8912 | . . . . . . 7 ⊢ (𝑝 = 𝑤 → sup(𝑝, 𝑉, 𝑅) = sup(𝑤, 𝑉, 𝑅)) | |
7 | 5, 6 | opeq12d 4814 | . . . . . 6 ⊢ (𝑝 = 𝑤 → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)〉) |
8 | 7 | cbvmptv 5172 | . . . . 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 2847 | . . . 4 ⊢ 𝐹 = (𝑤 ∈ 𝑃 ↦ 〈inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)〉) |
10 | 3, 9 | fmptd 6881 | . . 3 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃⟶𝑂) |
11 | 3ancomb 1095 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)) | |
12 | 3anass 1091 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))) | |
13 | 11, 12 | bitri 277 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))) |
14 | 1, 2, 4 | prproropf1olem4 43675 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
15 | 13, 14 | sylbir 237 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
16 | 15 | ralrimivva 3194 | . . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
17 | dff13 7016 | . . 3 ⊢ (𝐹:𝑃–1-1→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))) | |
18 | 10, 16, 17 | sylanbrc 585 | . 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1→𝑂) |
19 | 1, 2 | prproropf1olem1 43672 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → {(1st ‘𝑤), (2nd ‘𝑤)} ∈ 𝑃) |
20 | fveq2 6673 | . . . . . . 7 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝐹‘𝑧) = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})) | |
21 | 20 | eqeq2d 2835 | . . . . . 6 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))) |
22 | 21 | adantl 484 | . . . . 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 43674 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}) = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
24 | 1 | prproropf1olem0 43671 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝑂 ↔ (𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∧ ((1st ‘𝑤) ∈ 𝑉 ∧ (2nd ‘𝑤) ∈ 𝑉) ∧ (1st ‘𝑤)𝑅(2nd ‘𝑤))) |
25 | 24 | simp1bi 1141 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝑂 → 𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
26 | 25 | eqcomd 2830 | . . . . . . 7 ⊢ (𝑤 ∈ 𝑂 → 〈(1st ‘𝑤), (2nd ‘𝑤)〉 = 𝑤) |
27 | 26 | adantl 484 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 〈(1st ‘𝑤), (2nd ‘𝑤)〉 = 𝑤) |
28 | 23, 27 | eqtr2d 2860 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})) |
29 | 19, 22, 28 | rspcedvd 3629 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧)) |
30 | 29 | ralrimiva 3185 | . . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧)) |
31 | dffo3 6871 | . . 3 ⊢ (𝐹:𝑃–onto→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))) | |
32 | 10, 30, 31 | sylanbrc 585 | . 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–onto→𝑂) |
33 | df-f1o 6365 | . 2 ⊢ (𝐹:𝑃–1-1-onto→𝑂 ↔ (𝐹:𝑃–1-1→𝑂 ∧ 𝐹:𝑃–onto→𝑂)) | |
34 | 18, 32, 33 | sylanbrc 585 | 1 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 {crab 3145 ∩ cin 3938 𝒫 cpw 4542 {cpr 4572 〈cop 4576 class class class wbr 5069 ↦ cmpt 5149 Or wor 5476 × cxp 5556 ⟶wf 6354 –1-1→wf1 6355 –onto→wfo 6356 –1-1-onto→wf1o 6357 ‘cfv 6358 1st c1st 7690 2nd c2nd 7691 supcsup 8907 infcinf 8908 2c2 11695 ♯chash 13693 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: prproropen 43677 prproropreud 43678 |
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