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Mirrors > Home > MPE Home > Th. List > usgrac | Structured version Visualization version GIF version |
Description: An undirected simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.) |
Ref | Expression |
---|---|
usgrac | ⊢ (𝐺 ∈ USGrph → (1st ‘𝐺) USGrph (2nd ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgraop 25879 | . 2 ⊢ (𝐺 ∈ USGrph → ∃𝑣∃𝑒(𝐺 = 〈𝑣, 𝑒〉 ∧ 𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})) | |
2 | id 22 | . . . . . . . 8 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → 𝐺 = 〈𝑣, 𝑒〉) | |
3 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑣 ∈ V | |
4 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑒 ∈ V | |
5 | 3, 4 | op1std 7069 | . . . . . . . . 9 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → (1st ‘𝐺) = 𝑣) |
6 | 3, 4 | op2ndd 7070 | . . . . . . . . 9 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → (2nd ‘𝐺) = 𝑒) |
7 | 5, 6 | opeq12d 4348 | . . . . . . . 8 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → 〈(1st ‘𝐺), (2nd ‘𝐺)〉 = 〈𝑣, 𝑒〉) |
8 | 2, 7 | eqtr4d 2647 | . . . . . . 7 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) |
9 | 8 | eleq1d 2672 | . . . . . 6 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → (𝐺 ∈ USGrph ↔ 〈(1st ‘𝐺), (2nd ‘𝐺)〉 ∈ USGrph )) |
10 | df-br 4584 | . . . . . 6 ⊢ ((1st ‘𝐺) USGrph (2nd ‘𝐺) ↔ 〈(1st ‘𝐺), (2nd ‘𝐺)〉 ∈ USGrph ) | |
11 | 9, 10 | syl6bbr 277 | . . . . 5 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → (𝐺 ∈ USGrph ↔ (1st ‘𝐺) USGrph (2nd ‘𝐺))) |
12 | 11 | biimpd 218 | . . . 4 ⊢ (𝐺 = 〈𝑣, 𝑒〉 → (𝐺 ∈ USGrph → (1st ‘𝐺) USGrph (2nd ‘𝐺))) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝐺 = 〈𝑣, 𝑒〉 ∧ 𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}) → (𝐺 ∈ USGrph → (1st ‘𝐺) USGrph (2nd ‘𝐺))) |
14 | 13 | exlimivv 1847 | . 2 ⊢ (∃𝑣∃𝑒(𝐺 = 〈𝑣, 𝑒〉 ∧ 𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}) → (𝐺 ∈ USGrph → (1st ‘𝐺) USGrph (2nd ‘𝐺))) |
15 | 1, 14 | mpcom 37 | 1 ⊢ (𝐺 ∈ USGrph → (1st ‘𝐺) USGrph (2nd ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {crab 2900 𝒫 cpw 4108 〈cop 4131 class class class wbr 4583 dom cdm 5038 –1-1→wf1 5801 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 2c2 10947 #chash 12979 USGrph cusg 25859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-usgra 25862 |
This theorem is referenced by: edgprvtx 25914 usgrafiedg 25945 0eusgraiff0rgracl 26468 |
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