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Theorem suppvalbr 7419
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 suppval 7417 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
2 df-rab 3023 . . . 4 {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})}
3 vex 3307 . . . . . . 7 𝑥 ∈ V
43eldm 5428 . . . . . 6 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
5 df-sn 4286 . . . . . . . 8 {𝑍} = {𝑦𝑦 = 𝑍}
65neeq2i 2961 . . . . . . 7 ({𝑦𝑥𝑅𝑦} ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍})
7 imasng 5597 . . . . . . . . 9 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦})
83, 7ax-mp 5 . . . . . . . 8 (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦}
98neeq1i 2960 . . . . . . 7 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑍})
10 nabbi 2998 . . . . . . 7 (∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍) ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍})
116, 9, 103bitr4i 292 . . . . . 6 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
124, 11anbi12i 735 . . . . 5 ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1312abbii 2841 . . . 4 {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
142, 13eqtri 2746 . . 3 {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
1514a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))})
16 df-ne 2897 . . . . . . . 8 (𝑦𝑍 ↔ ¬ 𝑦 = 𝑍)
1716bicomi 214 . . . . . . 7 𝑦 = 𝑍𝑦𝑍)
1817bibi2i 326 . . . . . 6 ((𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍) ↔ (𝑥𝑅𝑦𝑦𝑍))
1918exbii 1887 . . . . 5 (∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
2019anbi2i 732 . . . 4 ((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
2120abbii 2841 . . 3 {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))}
2221a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
231, 15, 223eqtrd 2762 1 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1596  wex 1817  wcel 2103  {cab 2710  wne 2896  {crab 3018  Vcvv 3304  {csn 4285   class class class wbr 4760  dom cdm 5218  cima 5221  (class class class)co 6765   supp csupp 7415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-supp 7416
This theorem is referenced by:  suppimacnvss  7425  suppimacnv  7426
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