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Mirrors > Home > MPE Home > Th. List > gropd | Structured version Visualization version GIF version |
Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, 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) has this property. (Contributed by AV, 11-Oct-2020.) |
Ref | Expression |
---|---|
gropd.g | ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓)) |
gropd.v | ⊢ (𝜑 → 𝑉 ∈ 𝑈) |
gropd.e | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
Ref | Expression |
---|---|
gropd | ⊢ (𝜑 → [〈𝑉, 𝐸〉 / 𝑔]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5356 | . . 3 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, 𝐸〉 ∈ V) |
3 | gropd.g | . 2 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓)) | |
4 | gropd.v | . . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑈) | |
5 | gropd.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
6 | opvtxfv 26789 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
7 | opiedgfv 26792 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
8 | 6, 7 | jca 514 | . . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
9 | 4, 5, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
10 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑔〈𝑉, 𝐸〉 | |
11 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
12 | nfsbc1v 3792 | . . . 4 ⊢ Ⅎ𝑔[〈𝑉, 𝐸〉 / 𝑔]𝜓 | |
13 | 11, 12 | nfim 1897 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓) |
14 | fveqeq2 6679 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘〈𝑉, 𝐸〉) = 𝑉)) | |
15 | fveqeq2 6679 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) | |
16 | 14, 15 | anbi12d 632 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸))) |
17 | sbceq1a 3783 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (𝜓 ↔ [〈𝑉, 𝐸〉 / 𝑔]𝜓)) | |
18 | 16, 17 | imbi12d 347 | . . 3 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓))) |
19 | 10, 13, 18 | spcgf 3590 | . 2 ⊢ (〈𝑉, 𝐸〉 ∈ V → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓))) |
20 | 2, 3, 9, 19 | syl3c 66 | 1 ⊢ (𝜑 → [〈𝑉, 𝐸〉 / 𝑔]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3494 [wsbc 3772 〈cop 4573 ‘cfv 6355 Vtxcvtx 26781 iEdgciedg 26782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690 df-vtx 26783 df-iedg 26784 |
This theorem is referenced by: gropeld 26818 |
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