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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprsymrelf1o | Structured version Visualization version GIF version |
Description: The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
Ref | Expression |
---|---|
sprsymrelf.p | ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) |
sprsymrelf.r | ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} |
sprsymrelf.f | ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) |
Ref | Expression |
---|---|
sprsymrelf1o | ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sprsymrelf.p | . . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) | |
2 | sprsymrelf.r | . . . 4 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} | |
3 | sprsymrelf.f | . . . 4 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) | |
4 | 1, 2, 3 | sprsymrelf1 43651 | . . 3 ⊢ 𝐹:𝑃–1-1→𝑅 |
5 | 4 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1→𝑅) |
6 | 1, 2, 3 | sprsymrelfo 43652 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅) |
7 | df-f1o 6357 | . 2 ⊢ (𝐹:𝑃–1-1-onto→𝑅 ↔ (𝐹:𝑃–1-1→𝑅 ∧ 𝐹:𝑃–onto→𝑅)) | |
8 | 5, 6, 7 | sylanbrc 585 | 1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 𝒫 cpw 4539 {cpr 4563 class class class wbr 5059 {copab 5121 ↦ cmpt 5139 × cxp 5548 –1-1→wf1 6347 –onto→wfo 6348 –1-1-onto→wf1o 6349 ‘cfv 6350 Pairscspr 43632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-spr 43633 |
This theorem is referenced by: sprbisymrel 43654 uspgrbisymrelALT 44023 |
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