MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppvalbr Structured version   Visualization version   GIF version

Theorem suppvalbr 7244
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 7242 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
2 df-rab 2916 . . . 4 {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})}
3 vex 3189 . . . . . . 7 𝑥 ∈ V
43eldm 5281 . . . . . 6 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
5 df-sn 4149 . . . . . . . 8 {𝑍} = {𝑦𝑦 = 𝑍}
65neeq2i 2855 . . . . . . 7 ({𝑦𝑥𝑅𝑦} ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍})
7 imasng 5446 . . . . . . . . 9 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦})
83, 7ax-mp 5 . . . . . . . 8 (𝑅 “ {𝑥}) = {𝑦𝑥𝑅𝑦}
98neeq1i 2854 . . . . . . 7 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑍})
10 nabbi 2892 . . . . . . 7 (∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍) ↔ {𝑦𝑥𝑅𝑦} ≠ {𝑦𝑦 = 𝑍})
116, 9, 103bitr4i 292 . . . . . 6 ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
124, 11anbi12i 732 . . . . 5 ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
1312abbii 2736 . . . 4 {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
142, 13eqtri 2643 . . 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 2791 . . . . . . . 8 (𝑦𝑍 ↔ ¬ 𝑦 = 𝑍)
1716bicomi 214 . . . . . . 7 𝑦 = 𝑍𝑦𝑍)
1817bibi2i 327 . . . . . 6 ((𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍) ↔ (𝑥𝑅𝑦𝑦𝑍))
1918exbii 1771 . . . . 5 (∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
2019anbi2i 729 . . . 4 ((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
2120abbii 2736 . . 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 2659 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 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  {crab 2911  Vcvv 3186  {csn 4148   class class class wbr 4613  dom cdm 5074  cima 5077  (class class class)co 6604   supp csupp 7240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-supp 7241
This theorem is referenced by:  suppimacnvss  7250  suppimacnv  7251
  Copyright terms: Public domain W3C validator