Theorem usgraeq12d 25891

Description: Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)

Assertion

Ref	Expression
usgraeq12d	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝑉 USGrph 𝐸 ↔ 𝑊 USGrph 𝐹))

Proof of Theorem usgraeq12d

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 792	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → 𝐸 = 𝐹)
2		dmeq 5246	. . . . 5 ⊢ (𝐸 = 𝐹 → dom 𝐸 = dom 𝐹)
3	2	adantl 481	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
4	3	adantl 481	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → dom 𝐸 = dom 𝐹)
5		pweq 4111	. . . . . . 7 ⊢ (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
6	5	adantr 480	. . . . . 6 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝒫 𝑉 = 𝒫 𝑊)
7	6	adantl 481	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → 𝒫 𝑉 = 𝒫 𝑊)
8	7	difeq1d 3689	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
9	8	rabeqdv 3167	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2})
10	1, 4, 9	f1eq123d 6044	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐹:dom 𝐹–1-1→{𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2}))
11		isusgra 25873	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
12	11	adantr 480	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝑉 USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
13		eleq1 2676	. . . . . . . 8 ⊢ (𝑉 = 𝑊 → (𝑉 ∈ 𝑋 ↔ 𝑊 ∈ 𝑋))
14	13	biimpd 218	. . . . . . 7 ⊢ (𝑉 = 𝑊 → (𝑉 ∈ 𝑋 → 𝑊 ∈ 𝑋))
15	14	adantr 480	. . . . . 6 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝑉 ∈ 𝑋 → 𝑊 ∈ 𝑋))
16	15	com12 32	. . . . 5 ⊢ (𝑉 ∈ 𝑋 → ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝑊 ∈ 𝑋))
17	16	adantr 480	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝑊 ∈ 𝑋))
18	17	imp 444	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → 𝑊 ∈ 𝑋)
19		eleq1 2676	. . . . . . . 8 ⊢ (𝐸 = 𝐹 → (𝐸 ∈ 𝑌 ↔ 𝐹 ∈ 𝑌))
20	19	biimpd 218	. . . . . . 7 ⊢ (𝐸 = 𝐹 → (𝐸 ∈ 𝑌 → 𝐹 ∈ 𝑌))
21	20	adantl 481	. . . . . 6 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝐸 ∈ 𝑌 → 𝐹 ∈ 𝑌))
22	21	com12 32	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐹 ∈ 𝑌))
23	22	adantl 481	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐹 ∈ 𝑌))
24	23	imp 444	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → 𝐹 ∈ 𝑌)
25		isusgra 25873	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) → (𝑊 USGrph 𝐹 ↔ 𝐹:dom 𝐹–1-1→{𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2}))
26	18, 24, 25	syl2anc 691	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝑊 USGrph 𝐹 ↔ 𝐹:dom 𝐹–1-1→{𝑥 ∈ (𝒫 𝑊 ∖ {∅}) ∣ (#‘𝑥) = 2}))
27	10, 12, 26	3bitr4d 299	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝑉 USGrph 𝐸 ↔ 𝑊 USGrph 𝐹))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-usgra 25862

This theorem is referenced by: (None)