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

Theorem gropeld 26829
Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 11-Oct-2020.)
Hypotheses
Ref Expression
gropeld.g (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))
gropeld.v (𝜑𝑉𝑈)
gropeld.e (𝜑𝐸𝑊)
Assertion
Ref Expression
gropeld (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Distinct variable groups:   𝐶,𝑔   𝑔,𝐸   𝑔,𝑉   𝜑,𝑔
Allowed substitution hints:   𝑈(𝑔)   𝑊(𝑔)

Proof of Theorem gropeld
StepHypRef Expression
1 gropeld.g . . 3 (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))
2 gropeld.v . . 3 (𝜑𝑉𝑈)
3 gropeld.e . . 3 (𝜑𝐸𝑊)
41, 2, 3gropd 26827 . 2 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝑔𝐶)
5 sbcel1v 3789 . 2 ([𝑉, 𝐸⟩ / 𝑔]𝑔𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
64, 5sylib 221 1 (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wcel 2112  [wsbc 3723  cop 4534  cfv 6328  Vtxcvtx 26792  iEdgciedg 26793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7675  df-2nd 7676  df-vtx 26794  df-iedg 26795
This theorem is referenced by:  upgr0eopALT  26912  upgr1eopALT  26913  upgrspanop  27090  umgrspanop  27091  usgrspanop  27092  cplgrop  27230
  Copyright terms: Public domain W3C validator