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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 29048 | . 2 ⊢ (𝜑 → [〈𝑉, 𝐸〉 / 𝑔]𝑔 ∈ 𝐶) | 
| 5 | sbcel1v 3856 | . 2 ⊢ ([〈𝑉, 𝐸〉 / 𝑔]𝑔 ∈ 𝐶 ↔ 〈𝑉, 𝐸〉 ∈ 𝐶) | |
| 6 | 4, 5 | sylib 218 | 1 ⊢ (𝜑 → 〈𝑉, 𝐸〉 ∈ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 [wsbc 3788 〈cop 4632 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 df-vtx 29015 df-iedg 29016 | 
| This theorem is referenced by: upgr0eopALT 29133 upgr1eopALT 29134 upgrspanop 29314 umgrspanop 29315 usgrspanop 29316 cplgrop 29454 | 
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