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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 4859 | . . 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 25681 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
7 | opiedgfv 25684 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
8 | 6, 7 | jca 553 | . . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
9 | 4, 5, 8 | syl2anc 691 | . 2 ⊢ (𝜑 → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
10 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑔〈𝑉, 𝐸〉 | |
11 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
12 | nfsbc1v 3422 | . . . 4 ⊢ Ⅎ𝑔[〈𝑉, 𝐸〉 / 𝑔]𝜓 | |
13 | 11, 12 | nfim 1813 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓) |
14 | fveq2 6103 | . . . . . 6 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (Vtx‘𝑔) = (Vtx‘〈𝑉, 𝐸〉)) | |
15 | 14 | eqeq1d 2612 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘〈𝑉, 𝐸〉) = 𝑉)) |
16 | fveq2 6103 | . . . . . 6 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (iEdg‘𝑔) = (iEdg‘〈𝑉, 𝐸〉)) | |
17 | 16 | eqeq1d 2612 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
18 | 15, 17 | anbi12d 743 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸))) |
19 | sbceq1a 3413 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (𝜓 ↔ [〈𝑉, 𝐸〉 / 𝑔]𝜓)) | |
20 | 18, 19 | imbi12d 333 | . . 3 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓))) |
21 | 10, 13, 20 | spcgf 3261 | . 2 ⊢ (〈𝑉, 𝐸〉 ∈ V → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓))) |
22 | 2, 3, 9, 21 | syl3c 64 | 1 ⊢ (𝜑 → [〈𝑉, 𝐸〉 / 𝑔]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 〈cop 4131 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 df-vtx 25675 df-iedg 25676 |
This theorem is referenced by: gropeld 25710 |
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