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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 44046. (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 6670 | . 2 ⊢ (𝑔 = 𝑋 → (2nd ‘𝑔) = (2nd ‘𝑋)) | |
3 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐺 → 𝑋 ∈ 𝐺) | |
4 | fvexd 6685 | . 2 ⊢ (𝑋 ∈ 𝐺 → (2nd ‘𝑋) ∈ V) | |
5 | 1, 2, 3, 4 | fvmptd3 6791 | 1 ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 𝒫 cpw 4539 {copab 5128 ↦ cmpt 5146 ‘cfv 6355 2nd c2nd 7688 Vtxcvtx 26781 Edgcedg 26832 USPGraphcuspgr 26933 Pairscspr 43659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 |
This theorem is referenced by: uspgrsprf1 44042 uspgrsprfo 44043 |
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