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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 5363 | . . 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 27119 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
7 | opiedgfv 27122 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
8 | 6, 7 | jca 515 | . . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
9 | 4, 5, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) |
10 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑔〈𝑉, 𝐸〉 | |
11 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
12 | nfsbc1v 3729 | . . . 4 ⊢ Ⅎ𝑔[〈𝑉, 𝐸〉 / 𝑔]𝜓 | |
13 | 11, 12 | nfim 1904 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓) |
14 | fveqeq2 6745 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘〈𝑉, 𝐸〉) = 𝑉)) | |
15 | fveqeq2 6745 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) | |
16 | 14, 15 | anbi12d 634 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸))) |
17 | sbceq1a 3720 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (𝜓 ↔ [〈𝑉, 𝐸〉 / 𝑔]𝜓)) | |
18 | 16, 17 | imbi12d 348 | . . 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 3519 | . 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 399 ∀wal 1541 = wceq 1543 ∈ wcel 2111 Vcvv 3421 [wsbc 3709 〈cop 4562 ‘cfv 6398 Vtxcvtx 27111 iEdgciedg 27112 |
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 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-iota 6356 df-fun 6400 df-fv 6406 df-1st 7780 df-2nd 7781 df-vtx 27113 df-iedg 27114 |
This theorem is referenced by: gropeld 27148 |
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