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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprsymrelfv | Structured version Visualization version GIF version |
Description: The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-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 |
---|---|
sprsymrelfv | ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sprsymrelf.f | . . 3 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})) |
3 | rexeq 3322 | . . . 4 ⊢ (𝑝 = 𝑋 → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦})) | |
4 | 3 | opabbidv 4909 | . . 3 ⊢ (𝑝 = 𝑋 → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) |
5 | 4 | adantl 474 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑝 = 𝑋) → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) |
6 | id 22 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃) | |
7 | elpwi 4359 | . . . 4 ⊢ (𝑋 ∈ 𝒫 (Pairs‘𝑉) → 𝑋 ⊆ (Pairs‘𝑉)) | |
8 | sprsymrelf.p | . . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) | |
9 | 7, 8 | eleq2s 2896 | . . 3 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ⊆ (Pairs‘𝑉)) |
10 | sprsymrelfvlem 42539 | . . 3 ⊢ (𝑋 ⊆ (Pairs‘𝑉) → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑋 ∈ 𝑃 → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) |
12 | 2, 5, 6, 11 | fvmptd 6513 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∃wrex 3090 {crab 3093 ⊆ wss 3769 𝒫 cpw 4349 {cpr 4370 class class class wbr 4843 {copab 4905 ↦ cmpt 4922 × cxp 5310 ‘cfv 6101 Pairscspr 42526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-spr 42527 |
This theorem is referenced by: sprsymrelf1 42545 sprsymrelfo 42546 |
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