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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sprbisymrel | Structured version Visualization version GIF version | ||
| Description: There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and 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)} | 
| Ref | Expression | 
|---|---|
| sprbisymrel | ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sprsymrelf.p | . . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) | |
| 2 | fvex 6919 | . . . . 5 ⊢ (Pairs‘𝑉) ∈ V | |
| 3 | 2 | pwex 5380 | . . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V | 
| 4 | 1, 3 | eqeltri 2837 | . . 3 ⊢ 𝑃 ∈ V | 
| 5 | mptexg 7241 | . . 3 ⊢ (𝑃 ∈ V → (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V) | |
| 6 | 4, 5 | mp1i 13 | . 2 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V) | 
| 7 | sprsymrelf.r | . . 3 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} | |
| 8 | eqid 2737 | . . 3 ⊢ (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) | |
| 9 | 1, 7, 8 | sprsymrelf1o 47485 | . 2 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅) | 
| 10 | f1oeq1 6836 | . . 3 ⊢ (𝑓 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) → (𝑓:𝑃–1-1-onto→𝑅 ↔ (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅)) | |
| 11 | 10 | spcegv 3597 | . 2 ⊢ ((𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V → ((𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅)) | 
| 12 | 6, 9, 11 | sylc 65 | 1 ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 𝒫 cpw 4600 {cpr 4628 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 × cxp 5683 –1-1-onto→wf1o 6560 ‘cfv 6561 Pairscspr 47464 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-spr 47465 | 
| This theorem is referenced by: sprsymrelen 47487 | 
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