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Theorem suppvalbr 8189
Description: The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
Assertion
Ref Expression
suppvalbr ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑍,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem suppvalbr
StepHypRef Expression
1 df-rab 3437 . . . 4 {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})}
2 vex 3484 . . . . . . 7 𝑥 ∈ V
32eldm 5911 . . . . . 6 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4 imasng 6102 . . . . . . . . 9 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦})
54elv 3485 . . . . . . . 8 (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦}
65neeq1i 3005 . . . . . . 7 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑍})
7 df-sn 4627 . . . . . . . 8 {𝑍} = {𝑦𝑦 = 𝑍}
87neeq2i 3006 . . . . . . 7 ({𝑦𝑥𝑅𝑦} ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍})
9 nabbib 3045 . . . . . . 7 ({𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
106, 8, 93bitri 297 . . . . . 6 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
113, 10anbi12i 628 . . . . 5 ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1211abbii 2809 . . . 4 {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
131, 12eqtr2i 2766 . . 3 {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}}
1413a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
15 df-ne 2941 . . . . . . 7 (𝑦𝑍 ↔ ¬ 𝑦 = 𝑍)
1615bibi2i 337 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) ↔ (𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
1716exbii 1848 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
1817anbi2i 623 . . . 4 ((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1918abbii 2809 . . 3 {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
2019a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))})
21 suppval 8187 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
2214, 20, 213eqtr4rd 2788 1 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  {crab 3436  Vcvv 3480  {csn 4626   class class class wbr 5143  dom cdm 5685  cima 5688  (class class class)co 7431   supp csupp 8185
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-sep 5296  ax-nul 5306  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-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  suppimacnvss  8198  suppimacnv  8199
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