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Theorem sprsymrelfv 42223
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.)
Hypotheses
Ref Expression
sprsymrelf.p 𝑃 = 𝒫 (Pairs‘𝑉)
sprsymrelf.r 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
sprsymrelf.f 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
Assertion
Ref Expression
sprsymrelfv (𝑋𝑃 → (𝐹𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})
Distinct variable groups:   𝑃,𝑝   𝑉,𝑐,𝑥,𝑦   𝑝,𝑐,𝑥,𝑦,𝑋
Allowed substitution hints:   𝑃(𝑥,𝑦,𝑟,𝑐)   𝑅(𝑥,𝑦,𝑟,𝑝,𝑐)   𝐹(𝑥,𝑦,𝑟,𝑝,𝑐)   𝑉(𝑟,𝑝)   𝑋(𝑟)

Proof of Theorem sprsymrelfv
StepHypRef Expression
1 sprsymrelf.f . . 3 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
21a1i 11 . 2 (𝑋𝑃𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}))
3 rexeq 3266 . . . 4 (𝑝 = 𝑋 → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}))
43opabbidv 4856 . . 3 (𝑝 = 𝑋 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})
54adantl 473 . 2 ((𝑋𝑃𝑝 = 𝑋) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})
6 id 22 . 2 (𝑋𝑃𝑋𝑃)
7 elpwi 4300 . . . 4 (𝑋 ∈ 𝒫 (Pairs‘𝑉) → 𝑋 ⊆ (Pairs‘𝑉))
8 sprsymrelf.p . . . 4 𝑃 = 𝒫 (Pairs‘𝑉)
97, 8eleq2s 2845 . . 3 (𝑋𝑃𝑋 ⊆ (Pairs‘𝑉))
10 sprsymrelfvlem 42219 . . 3 (𝑋 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
119, 10syl 17 . 2 (𝑋𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
122, 5, 6, 11fvmptd 6438 1 (𝑋𝑃 → (𝐹𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  wcel 2127  wral 3038  wrex 3039  {crab 3042  wss 3703  𝒫 cpw 4290  {cpr 4311   class class class wbr 4792  {copab 4852  cmpt 4869   × cxp 5252  cfv 6037  Pairscspr 42206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-spr 42207
This theorem is referenced by:  sprsymrelf1  42225  sprsymrelfo  42226
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