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| 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 5469 | . . 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 29021 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
| 7 | opiedgfv 29024 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
| 8 | 6, 7 | jca 511 | . . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) | 
| 9 | 4, 5, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → ((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸)) | 
| 10 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑔〈𝑉, 𝐸〉 | |
| 11 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
| 12 | nfsbc1v 3808 | . . . 4 ⊢ Ⅎ𝑔[〈𝑉, 𝐸〉 / 𝑔]𝜓 | |
| 13 | 11, 12 | nfim 1896 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘〈𝑉, 𝐸〉) = 𝑉 ∧ (iEdg‘〈𝑉, 𝐸〉) = 𝐸) → [〈𝑉, 𝐸〉 / 𝑔]𝜓) | 
| 14 | fveqeq2 6915 | . . . . 5 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘〈𝑉, 𝐸〉) = 𝑉)) | |
| 15 | fveqeq2 6915 | . . . . 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 3799 | . . . 4 ⊢ (𝑔 = 〈𝑉, 𝐸〉 → (𝜓 ↔ [〈𝑉, 𝐸〉 / 𝑔]𝜓)) | |
| 18 | 16, 17 | imbi12d 344 | . . 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 3591 | . 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 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [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: gropeld 29050 | 
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