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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 46749 | . . 3 ⊢ 𝐹:𝑃–1-1→𝑅 |
5 | 4 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1→𝑅) |
6 | 1, 2, 3 | sprsymrelfo 46750 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅) |
7 | df-f1o 6549 | . 2 ⊢ (𝐹:𝑃–1-1-onto→𝑅 ↔ (𝐹:𝑃–1-1→𝑅 ∧ 𝐹:𝑃–onto→𝑅)) | |
8 | 5, 6, 7 | sylanbrc 582 | 1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {crab 3427 𝒫 cpw 4598 {cpr 4626 class class class wbr 5142 {copab 5204 ↦ cmpt 5225 × cxp 5670 –1-1→wf1 6539 –onto→wfo 6540 –1-1-onto→wf1o 6541 ‘cfv 6542 Pairscspr 46730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-spr 46731 |
This theorem is referenced by: sprbisymrel 46752 uspgrbisymrelALT 47130 |
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