Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgrexi | Structured version Visualization version GIF version |
Description: An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) |
Ref | Expression |
---|---|
usgrexi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} |
Ref | Expression |
---|---|
usgrexi | ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6086 | . . . . . 6 ⊢ ( I ↾ 𝑃):𝑃–1-1-onto→𝑃 | |
2 | f1of1 6049 | . . . . . 6 ⊢ (( I ↾ 𝑃):𝑃–1-1-onto→𝑃 → ( I ↾ 𝑃):𝑃–1-1→𝑃) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝑃):𝑃–1-1→𝑃 |
4 | dmresi 5376 | . . . . . 6 ⊢ dom ( I ↾ 𝑃) = 𝑃 | |
5 | f1eq2 6010 | . . . . . 6 ⊢ (dom ( I ↾ 𝑃) = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 ↔ ( I ↾ 𝑃):𝑃–1-1→𝑃)) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 ↔ ( I ↾ 𝑃):𝑃–1-1→𝑃) |
7 | 3, 6 | mpbir 220 | . . . 4 ⊢ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 |
8 | usgrexi.p | . . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} | |
9 | 8 | eqcomi 2619 | . . . . 5 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = 𝑃 |
10 | f1eq3 6011 | . . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃)) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃)) |
12 | 7, 11 | mpbiri 247 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) |
13 | pwexg 4776 | . . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V) | |
14 | 8, 13 | rabexd 4741 | . . . . . 6 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) |
15 | resiexg 6994 | . . . . . 6 ⊢ (𝑃 ∈ V → ( I ↾ 𝑃) ∈ V) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V) |
17 | opiedgfv 25684 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = ( I ↾ 𝑃)) | |
18 | 16, 17 | mpdan 699 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = ( I ↾ 𝑃)) |
19 | 18 | dmeqd 5248 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = dom ( I ↾ 𝑃)) |
20 | opvtxfv 25681 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝑉) | |
21 | 16, 20 | mpdan 699 | . . . . . 6 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝑉) |
22 | 21 | pweqd 4113 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝒫 𝑉) |
23 | 22 | rabeqdv 3167 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) |
24 | 18, 19, 23 | f1eq123d 6044 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ((iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})) |
25 | 12, 24 | mpbird 246 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (#‘𝑥) = 2}) |
26 | opex 4859 | . . 3 ⊢ 〈𝑉, ( I ↾ 𝑃)〉 ∈ V | |
27 | eqid 2610 | . . . 4 ⊢ (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) | |
28 | eqid 2610 | . . . 4 ⊢ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) | |
29 | 27, 28 | isusgrs 40386 | . . 3 ⊢ (〈𝑉, ( I ↾ 𝑃)〉 ∈ V → (〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph ↔ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (#‘𝑥) = 2})) |
30 | 26, 29 | mp1i 13 | . 2 ⊢ (𝑉 ∈ 𝑊 → (〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph ↔ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (#‘𝑥) = 2})) |
31 | 25, 30 | mpbird 246 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 𝒫 cpw 4108 〈cop 4131 I cid 4948 dom cdm 5038 ↾ cres 5040 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 USGraph cusgr 40379 |
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-vtx 25675 df-iedg 25676 df-usgr 40381 |
This theorem is referenced by: cusgrexi 40662 |
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