wrdumgren - Intuitionistic Logic Explorer

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Theorem wrdumgren 16027

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 16026	. . 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 11166	. . . . 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 5496	. . 3 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → dom 𝐸 = (0..^(♯‘𝐸)))
8	7	feq2d 5477	. 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 11164	. . . . . 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 11165	. . . . 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 11166	. . 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 2202 {crab 2515 𝒫 cpw 3656 class class class wbr 4093 dom cdm 4731 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 2_oc2o 6619 ≈ cen 6950 0cc0 8075 ℕ₀cn0 9445 ..^cfzo 10420 ♯chash 11081 Word cword 11160 Vtxcvtx 15933 iEdgciedg 15934 UMGraphcumgr 16013

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-1o 6625 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-5 9248 df-6 9249 df-7 9250 df-8 9251 df-9 9252 df-n0 9446 df-z 9523 df-dec 9655 df-uz 9799 df-fz 10287 df-fzo 10421 df-ihash 11082 df-word 11161 df-ndx 13146 df-slot 13147 df-base 13149 df-edgf 15926 df-vtx 15935 df-iedg 15936 df-umgren 16015

This theorem is referenced by: konigsbergumgr 16408

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