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Mirrors > Home > MPE Home > Th. List > upgr0eopALT | Structured version Visualization version GIF version |
Description: Alternate proof of upgr0eop 26229, using the general theorem gropeld 26145 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 26229). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
upgr0eopALT | ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3352 | . . . . . 6 ⊢ 𝑔 ∈ V | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V) |
3 | simpr 471 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅) | |
4 | 2, 3 | upgr0e 26226 | . . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph) |
5 | 4 | ax-gen 1869 | . . 3 ⊢ ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph) |
6 | 5 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)) |
7 | id 22 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ 𝑊) | |
8 | 0ex 4921 | . . 3 ⊢ ∅ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V) |
10 | 6, 7, 9 | gropeld 26145 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∀wal 1628 = wceq 1630 ∈ wcel 2144 Vcvv 3349 ∅c0 4061 〈cop 4320 ‘cfv 6031 Vtxcvtx 26094 iEdgciedg 26095 UPGraphcupgr 26195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-i2m1 10205 ax-1ne0 10206 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-1st 7314 df-2nd 7315 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-2 11280 df-vtx 26096 df-iedg 26097 df-upgr 26197 df-umgr 26198 |
This theorem is referenced by: (None) |
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