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Mirrors > Home > MPE Home > Th. List > gropeld | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
gropeld.g | ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔 ∈ 𝐶)) |
gropeld.v | ⊢ (𝜑 → 𝑉 ∈ 𝑈) |
gropeld.e | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
Ref | Expression |
---|---|
gropeld | ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gropeld.g | . . 3 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔 ∈ 𝐶)) | |
2 | gropeld.v | . . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑈) | |
3 | gropeld.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
4 | 1, 2, 3 | gropd 28864 | . 2 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶) |
5 | sbcel1v 3849 | . 2 ⊢ ([⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶) | |
6 | 4, 5 | sylib 217 | 1 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 [wsbc 3778 ⟨cop 4638 ‘cfv 6553 Vtxcvtx 28829 iEdgciedg 28830 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fv 6561 df-1st 7999 df-2nd 8000 df-vtx 28831 df-iedg 28832 |
This theorem is referenced by: upgr0eopALT 28949 upgr1eopALT 28950 upgrspanop 29130 umgrspanop 29131 usgrspanop 29132 cplgrop 29270 |
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