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Theorem dfssr2 35726
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 5462 . . . . . . 7 (𝑧 E 𝑥𝑧𝑥)
2 brvdif 35509 . . . . . . . 8 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦)
3 epel 5462 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
42, 3xchbinx 336 . . . . . . 7 (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧𝑦)
51, 4anbi12i 628 . . . . . 6 ((𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑦))
65exbii 1841 . . . . 5 (∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
76notbii 322 . . . 4 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
8 dfss6 3955 . . . 4 (𝑥𝑦 ↔ ¬ ∃𝑧(𝑧𝑥 ∧ ¬ 𝑧𝑦))
97, 8bitr4i 280 . . 3 (¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦) ↔ 𝑥𝑦)
109opabbii 5124 . 2 {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
11 rnxrn 35633 . . . 4 ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1211difeq2i 4094 . . 3 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)})
13 vvdifopab 35508 . . 3 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
1412, 13eqtri 2842 . 2 ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥𝑧(V ∖ E )𝑦)}
15 df-ssr 35725 . 2 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1610, 14, 153eqtr4ri 2853 1 S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 398   = wceq 1530  wex 1773  wcel 2107  Vcvv 3493  cdif 3931  wss 3934   class class class wbr 5057  {copab 5119   E cep 5457   × cxp 5546  ran crn 5549  cxrn 35439   S cssr 35443
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7681  df-2nd 7682  df-ec 8283  df-xrn 35610  df-ssr 35725
This theorem is referenced by: (None)
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