wrdumgren - Intuitionistic Logic Explorer

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Theorem wrdumgren 16213

Description: The property of being an undirected multigraph, expressing the edges as "words". (Contributed by AV, 24-Nov-2020.)

**Hypotheses**
Ref	Expression
isumgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isumgr.e	⊢ 𝐸 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
wrdumgren	⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))

Distinct variable groups: 𝑥,𝐺 𝑥,𝑉 Allowed substitution hints: 𝑈(𝑥) 𝐸(𝑥) 𝑋(𝑥)

**Proof of Theorem wrdumgren**
Step	Hyp	Ref	Expression
1		isumgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		isumgr.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isumgren 16212	. . 3 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))
4	3	adantr 276	. 2 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))
5		wrdf 11255	. . . . 5 ⊢ (𝐸 ∈ Word 𝑋 → 𝐸:(0..^(♯‘𝐸))⟶𝑋)
6	5	adantl 277	. . . 4 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → 𝐸:(0..^(♯‘𝐸))⟶𝑋)
7	6	fdmd 5520	. . 3 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → dom 𝐸 = (0..^(♯‘𝐸)))
8	7	feq2d 5501	. 2 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} ↔ 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))
9		simpr 110	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) ∧ 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}) → 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o})
10		lencl 11253	. . . . . 6 ⊢ (𝐸 ∈ Word 𝑋 → (♯‘𝐸) ∈ ℕ₀)
11	10	ad2antlr 489	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) ∧ 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}) → (♯‘𝐸) ∈ ℕ₀)
12		iswrdinn0 11254	. . . . 5 ⊢ ((𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} ∧ (♯‘𝐸) ∈ ℕ₀) → 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o})
13	9, 11, 12	syl2anc 411	. . . 4 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) ∧ 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}) → 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o})
14	13	ex 115	. . 3 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} → 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))
15		wrdf 11255	. . 3 ⊢ (𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} → 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o})
16	14, 15	impbid1 142	. 2 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o} ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))
17	4, 8, 16	3bitrd 214	1 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2205 {crab 2526 𝒫 cpw 3674 class class class wbr 4114 dom cdm 4754 ⟶wf 5353 ‘cfv 5357 (class class class)co 6058 2_oc2o 6654 ≈ cen 6986 0cc0 8143 ℕ₀cn0 9513 ..^cfzo 10498 ♯chash 11163 Word cword 11249 Vtxcvtx 16119 iEdgciedg 16120 UMGraphcumgr 16199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-dec 9728 df-uz 9872 df-fz 10362 df-fzo 10499 df-ihash 11164 df-word 11250 df-ndx 13299 df-slot 13300 df-base 13302 df-edgf 16112 df-vtx 16121 df-iedg 16122 df-umgren 16201

This theorem is referenced by: konigsbergumgr 16594

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