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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 28794 | . 2 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶) |
5 | sbcel1v 3843 | . 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 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 [wsbc 3772 ⟨cop 4629 ‘cfv 6536 Vtxcvtx 28759 iEdgciedg 28760 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-1st 7971 df-2nd 7972 df-vtx 28761 df-iedg 28762 |
This theorem is referenced by: upgr0eopALT 28879 upgr1eopALT 28880 upgrspanop 29057 umgrspanop 29058 usgrspanop 29059 cplgrop 29197 |
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