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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 44636 | . . 3 ⊢ 𝐹:𝑃–1-1→𝑅 |
5 | 4 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1→𝑅) |
6 | 1, 2, 3 | sprsymrelfo 44637 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅) |
7 | df-f1o 6396 | . 2 ⊢ (𝐹:𝑃–1-1-onto→𝑅 ↔ (𝐹:𝑃–1-1→𝑅 ∧ 𝐹:𝑃–onto→𝑅)) | |
8 | 5, 6, 7 | sylanbrc 586 | 1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ∃wrex 3063 {crab 3066 𝒫 cpw 4522 {cpr 4552 class class class wbr 5062 {copab 5124 ↦ cmpt 5144 × cxp 5558 –1-1→wf1 6386 –onto→wfo 6387 –1-1-onto→wf1o 6388 ‘cfv 6389 Pairscspr 44617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-spr 44618 |
This theorem is referenced by: sprbisymrel 44639 uspgrbisymrelALT 45005 |
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