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Theorem suppvalbr 8149
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 3433 . . . 4 {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})}
2 vex 3478 . . . . . . 7 𝑥 ∈ V
32eldm 5900 . . . . . 6 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4 imasng 6082 . . . . . . . . 9 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦})
54elv 3480 . . . . . . . 8 (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦}
65neeq1i 3005 . . . . . . 7 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑍})
7 df-sn 4629 . . . . . . . 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 296 . . . . . 6 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
113, 10anbi12i 627 . . . . 5 ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1211abbii 2802 . . . 4 {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
131, 12eqtr2i 2761 . . 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 1850 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
1817anbi2i 623 . . . 4 ((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1918abbii 2802 . . 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 8147 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
2214, 20, 213eqtr4rd 2783 1 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wne 2940  {crab 3432  Vcvv 3474  {csn 4628   class class class wbr 5148  dom cdm 5676  cima 5679  (class class class)co 7408   supp csupp 8145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-supp 8146
This theorem is referenced by:  suppimacnvss  8157  suppimacnv  8158
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