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Mathbox for Alexander van der Vekens |
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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 | . 2 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) | |
2 | rexeq 3322 | . . 3 ⊢ (𝑝 = 𝑋 → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦})) | |
3 | 2 | opabbidv 5214 | . 2 ⊢ (𝑝 = 𝑋 → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) |
4 | id 22 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃) | |
5 | elpwi 4609 | . . . 4 ⊢ (𝑋 ∈ 𝒫 (Pairs‘𝑉) → 𝑋 ⊆ (Pairs‘𝑉)) | |
6 | sprsymrelf.p | . . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) | |
7 | 5, 6 | eleq2s 2852 | . . 3 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ⊆ (Pairs‘𝑉)) |
8 | sprsymrelfvlem 46145 | . . 3 ⊢ (𝑋 ⊆ (Pairs‘𝑉) → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑋 ∈ 𝑃 → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) |
10 | 1, 3, 4, 9 | fvmptd3 7019 | 1 ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ⊆ wss 3948 𝒫 cpw 4602 {cpr 4630 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 × cxp 5674 ‘cfv 6541 Pairscspr 46132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6493 df-fun 6543 df-fv 6549 df-spr 46133 |
This theorem is referenced by: sprsymrelf1 46151 sprsymrelfo 46152 |
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