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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 27076 | . 2 ⊢ (𝜑 → [〈𝑉, 𝐸〉 / 𝑔]𝑔 ∈ 𝐶) |
5 | sbcel1v 3753 | . 2 ⊢ ([〈𝑉, 𝐸〉 / 𝑔]𝑔 ∈ 𝐶 ↔ 〈𝑉, 𝐸〉 ∈ 𝐶) | |
6 | 4, 5 | sylib 221 | 1 ⊢ (𝜑 → 〈𝑉, 𝐸〉 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 [wsbc 3683 〈cop 4533 ‘cfv 6358 Vtxcvtx 27041 iEdgciedg 27042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-1st 7739 df-2nd 7740 df-vtx 27043 df-iedg 27044 |
This theorem is referenced by: upgr0eopALT 27161 upgr1eopALT 27162 upgrspanop 27339 umgrspanop 27340 usgrspanop 27341 cplgrop 27479 |
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