Theorem isubgr0uhgr 48459

Description: The subgraph induced by an empty set of vertices of a hypergraph. (Contributed by AV, 13-May-2025.)

Assertion

Ref	Expression
isubgr0uhgr	⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

Proof of Theorem isubgr0uhgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4353	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		eqid 2761	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2761	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	isisubgr 48448	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ∅ ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr ∅) = ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩)
5	1, 4	mpan2 701	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩)
6		inrab2 4269	. . . . . . 7 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
7		inidm 4178	. . . . . . . . 9 ⊢ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) = dom (iEdg‘𝐺)
8	7	rabeqi 3426	. . . . . . . 8 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
9		ss0b 4354	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) ⊆ ∅ ↔ ((iEdg‘𝐺)‘𝑥) = ∅)
10	8, 9	rabbieq 3421	. . . . . . 7 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
11	6, 10	eqtri 2784	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
12	11	ineqcomi 4163	. . . . 5 ⊢ (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
13	2, 3	uhgrf 29209	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
14		ffvelcdm 7058	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
15		eldifsni 4749	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((iEdg‘𝐺)‘𝑥) ≠ ∅)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ≠ ∅)
17	16	neneqd 2961	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
18	13, 17	sylan 589	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
19	18	ralrimiva 3153	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
20		rabeq0 4341	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
21	19, 20	sylibr 236	. . . . 5 ⊢ (𝐺 ∈ UHGraph → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅} = ∅)
22	12, 21	eqtrid 2808	. . . 4 ⊢ (𝐺 ∈ UHGraph → (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
23	3	uhgrfun 29213	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	funfnd 6548	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
25		fnresdisj 6637	. . . . 5 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → ((dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅ ↔ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅))
26	24, 25	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅ ↔ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅))
27	22, 26	mpbid 234	. . 3 ⊢ (𝐺 ∈ UHGraph → ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
28	27	opeq2d 4837	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩ = ⟨∅, ∅⟩)
29	5, 28	eqtrd 2796	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  {crab 3413   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  {csn 4581  ⟨cop 4587  dom cdm 5645   ↾ cres 5647   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  Vtxcvtx 29143  iEdgciedg 29144  UHGraphcuhgr 29203   ISubGr cisubgr 48446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-uhgr 29205  df-isubgr 48447

This theorem is referenced by: (None)