Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfssr2 Structured version   Visualization version   GIF version

Theorem dfssr2 37863
Description: Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
Assertion
Ref Expression
dfssr2 S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))

Proof of Theorem dfssr2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epel 5574 . . . . . . 7 (𝑧 E 𝑥𝑧𝑥)
2 brvdif 37623 . . . . . . . 8 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3 epel 5574 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
42, 3xchbinx 334 . . . . . . 7 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧𝑦)
51, 4anbi12i 626 . . . . . 6 ((𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑦))
65exbii 1842 . . . . 5 (∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
76notbii 320 . . . 4 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
8 dfss6 3964 . . . 4 (𝑥𝑦 ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
97, 8bitr4i 278 . . 3 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ 𝑥𝑦)
109opabbii 5206 . 2 {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
11 rnxrn 37762 . . . 4 ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1211difeq2i 4112 . . 3 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)})
13 vvdifopab 37622 . . 3 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1412, 13eqtri 2752 . 2 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
15 df-ssr 37862 . 2 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1610, 14, 153eqtr4ri 2763 1 S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1533  wex 1773  wcel 2098  Vcvv 3466  cdif 3938  wss 3941   class class class wbr 5139  {copab 5201   E cep 5570   × cxp 5665  ran crn 5668  cxrn 37536   S cssr 37540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-eprel 5571  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-1st 7969  df-2nd 7970  df-ec 8702  df-xrn 37735  df-ssr 37862
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator