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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 44379. (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 6645 | . 2 ⊢ (𝑔 = 𝑋 → (2nd ‘𝑔) = (2nd ‘𝑋)) | |
3 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐺 → 𝑋 ∈ 𝐺) | |
4 | fvexd 6660 | . 2 ⊢ (𝑋 ∈ 𝐺 → (2nd ‘𝑋) ∈ V) | |
5 | 1, 2, 3, 4 | fvmptd3 6768 | 1 ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 𝒫 cpw 4497 {copab 5092 ↦ cmpt 5110 ‘cfv 6324 2nd c2nd 7670 Vtxcvtx 26789 Edgcedg 26840 USPGraphcuspgr 26941 Pairscspr 43994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: uspgrsprf1 44375 uspgrsprfo 44376 |
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