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Theorem suppvalbr 8168
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 3421 . . . 4 {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})}
2 vex 3467 . . . . . . 7 𝑥 ∈ V
32eldm 5898 . . . . . 6 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4 imasng 6084 . . . . . . . . 9 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦})
54elv 3469 . . . . . . . 8 (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦}
65neeq1i 2995 . . . . . . 7 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑍})
7 df-sn 4625 . . . . . . . 8 {𝑍} = {𝑦𝑦 = 𝑍}
87neeq2i 2996 . . . . . . 7 ({𝑦𝑥𝑅𝑦} ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍})
9 nabbib 3035 . . . . . . 7 ({𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
106, 8, 93bitri 296 . . . . . 6 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
113, 10anbi12i 626 . . . . 5 ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1211abbii 2796 . . . 4 {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
131, 12eqtr2i 2755 . . 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 2931 . . . . . . 7 (𝑦𝑍 ↔ ¬ 𝑦 = 𝑍)
1615bibi2i 336 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) ↔ (𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
1716exbii 1843 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
1817anbi2i 621 . . . 4 ((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1918abbii 2796 . . 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 8166 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
2214, 20, 213eqtr4rd 2777 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 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wne 2930  {crab 3420  Vcvv 3463  {csn 4624   class class class wbr 5144  dom cdm 5673  cima 5676  (class class class)co 7414   supp csupp 8164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-supp 8165
This theorem is referenced by:  suppimacnvss  8177  suppimacnv  8178
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