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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uspgrsprfv | Structured version Visualization version GIF version |
Description: The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 47912. (Contributed by AV, 24-Nov-2021.) |
Ref | Expression |
---|---|
uspgrsprf.p | ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) |
uspgrsprf.g | ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} |
uspgrsprf.f | ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) |
Ref | Expression |
---|---|
uspgrsprfv | ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrsprf.f | . 2 ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) | |
2 | fveq2 6901 | . 2 ⊢ (𝑔 = 𝑋 → (2nd ‘𝑔) = (2nd ‘𝑋)) | |
3 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐺 → 𝑋 ∈ 𝐺) | |
4 | fvexd 6916 | . 2 ⊢ (𝑋 ∈ 𝐺 → (2nd ‘𝑋) ∈ V) | |
5 | 1, 2, 3, 4 | fvmptd3 7033 | 1 ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∃wrex 3066 Vcvv 3477 𝒫 cpw 4604 {copab 5211 ↦ cmpt 5232 ‘cfv 6558 2nd c2nd 8006 Vtxcvtx 29009 Edgcedg 29060 USPGraphcuspgr 29161 Pairscspr 47352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6510 df-fun 6560 df-fv 6566 |
This theorem is referenced by: uspgrsprf1 47908 uspgrsprfo 47909 |
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