wrdumgren - Intuitionistic Logic Explorer

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Theorem wrdumgren 16359

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 16358	. . 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 11312	. . . . 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 5540	. . 3 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → dom 𝐸 = (0..^(♯‘𝐸)))
8	7	feq2d 5521	. 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 11310	. . . . . 6 ⊢ (𝐸 ∈ Word 𝑋 → (♯‘𝐸) ∈ ℕ₀)
11	10	ad2antlr 493	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) ∧ 𝐸:(0..^(♯‘𝐸))⟶{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}) → (♯‘𝐸) ∈ ℕ₀)
12		iswrdinn0 11311	. . . . 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 415	. . . 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 11312	. . 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 {crab 2532 𝒫 cpw 3688 class class class wbr 4130 dom cdm 4774 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 2_oc2o 6681 ≈ cen 7020 0cc0 8179 ℕ₀cn0 9565 ..^cfzo 10551 ♯chash 11216 Word cword 11306 Vtxcvtx 16265 iEdgciedg 16266 UMGraphcumgr 16345

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-5 9367 df-6 9368 df-7 9369 df-8 9370 df-9 9371 df-n0 9566 df-z 9647 df-dec 9780 df-uz 9924 df-fz 10414 df-fzo 10552 df-ihash 11217 df-word 11307 df-ndx 13357 df-slot 13358 df-base 13360 df-edgf 16258 df-vtx 16267 df-iedg 16268 df-umgren 16347

This theorem is used by: konigsbergumgr 16740

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