isubgrgrim - Mathbox for Alexander van der Vekens

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Theorem isubgrgrim 49242

Description: Isomorphic subgraphs induced by subsets of vertices of two graphs. (Contributed by AV, 29-May-2025.)

**Hypotheses**
Ref	Expression
isubgrgrim.v	⊢ 𝑉 = (Vtx‘𝐺)
isubgrgrim.w	⊢ 𝑊 = (Vtx‘𝐻)
isubgrgrim.i	⊢ 𝐼 = (iEdg‘𝐺)
isubgrgrim.j	⊢ 𝐽 = (iEdg‘𝐻)
isubgrgrim.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
isubgrgrim.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

**Assertion**
Ref	Expression
isubgrgrim	⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝐺 ISubGr 𝑁) ≃_𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ ∃𝑔(𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Distinct variable groups: 𝑓,𝐺,𝑔,𝑖 𝑥,𝐺 𝑓,𝐻,𝑔,𝑖 𝑥,𝐻 𝑥,𝐼 𝑥,𝐽 𝑓,𝑀,𝑔,𝑖 𝑥,𝑀 𝑓,𝑁,𝑔,𝑖 𝑥,𝑁 𝑇,𝑓,𝑔,𝑖 𝑈,𝑓,𝑔,𝑖 𝑓,𝑉,𝑔,𝑖 𝑥,𝑉 𝑓,𝑊,𝑔,𝑖 𝑥,𝑊 𝑖,𝐾 𝑖,𝐿 Allowed substitution hints: 𝑇(𝑥) 𝑈(𝑥) 𝐼(𝑓, 𝑔, 𝑖) 𝐽(𝑓, 𝑔, 𝑖) 𝐾(𝑥, 𝑓, 𝑔) 𝐿(𝑥, 𝑓, 𝑔)

**Proof of Theorem isubgrgrim**
Step	Hyp	Ref	Expression
1		ovex 7479	. . . 4 ⊢ (𝐺 ISubGr 𝑁) ∈ V
2		ovex 7479	. . . 4 ⊢ (𝐻 ISubGr 𝑀) ∈ V
3	1, 2	pm3.2i 476	. . 3 ⊢ ((𝐺 ISubGr 𝑁) ∈ V ∧ (𝐻 ISubGr 𝑀) ∈ V)
4		eqid 2763	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑁)) = (Vtx‘(𝐺 ISubGr 𝑁))
5		eqid 2763	. . . 4 ⊢ (Vtx‘(𝐻 ISubGr 𝑀)) = (Vtx‘(𝐻 ISubGr 𝑀))
6		eqid 2763	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑁)) = (iEdg‘(𝐺 ISubGr 𝑁))
7		eqid 2763	. . . 4 ⊢ (iEdg‘(𝐻 ISubGr 𝑀)) = (iEdg‘(𝐻 ISubGr 𝑀))
8	4, 5, 6, 7	dfgric2 49228	. . 3 ⊢ (((𝐺 ISubGr 𝑁) ∈ V ∧ (𝐻 ISubGr 𝑀) ∈ V) → ((𝐺 ISubGr 𝑁) ≃_𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝑁))⟶1-1onto(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))))))
9	3, 8	mp1i 14	. 2 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝐺 ISubGr 𝑁) ≃_𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝑁))⟶1-1onto(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))))))
10		eqidd 2764	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → 𝑓 = 𝑓)
11		isubgrgrim.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	11	isubgrvtx 49180	. . . . . 6 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑁 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑁)) = 𝑁)
13	12	ad2ant2r 760	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (Vtx‘(𝐺 ISubGr 𝑁)) = 𝑁)
14		isubgrgrim.w	. . . . . . 7 ⊢ 𝑊 = (Vtx‘𝐻)
15	14	isubgrvtx 49180	. . . . . 6 ⊢ ((𝐻 ∈ 𝑇 ∧ 𝑀 ⊆ 𝑊) → (Vtx‘(𝐻 ISubGr 𝑀)) = 𝑀)
16	15	ad2ant2l 759	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (Vtx‘(𝐻 ISubGr 𝑀)) = 𝑀)
17	10, 13, 16	f1oeq123d 6843	. . . 4 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑓:(Vtx‘(𝐺 ISubGr 𝑁))⟶1-1onto(Vtx‘(𝐻 ISubGr 𝑀)) ↔ 𝑓:𝑁⟶1-1onto𝑀))
18		eqidd 2764	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → 𝑔 = 𝑔)
19		isubgrgrim.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (iEdg‘𝐺)
20	11, 19	isubgriedg 49176	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑁 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑁)) = (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}))
21	20	ad2ant2r 760	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐺 ISubGr 𝑁)) = (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}))
22	21	dmeqd 5895	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐺 ISubGr 𝑁)) = dom (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}))
23		ssrab2 4008	. . . . . . . . . . 11 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} ⊆ dom 𝐼
24	23	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} ⊆ dom 𝐼)
25		ssdmres 6017	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} ⊆ dom 𝐼 ↔ dom (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}) = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁})
26	24, 25	sylib 221	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}) = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁})
27		isubgrgrim.k	. . . . . . . . . . 11 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
28	27	eqcomi 2772	. . . . . . . . . 10 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} = 𝐾
29	28	a1i 11	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} = 𝐾)
30	22, 26, 29	3eqtrd 2802	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐺 ISubGr 𝑁)) = 𝐾)
31		isubgrgrim.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (iEdg‘𝐻)
32	14, 31	isubgriedg 49176	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ 𝑇 ∧ 𝑀 ⊆ 𝑊) → (iEdg‘(𝐻 ISubGr 𝑀)) = (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}))
33	32	ad2ant2l 759	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐻 ISubGr 𝑀)) = (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}))
34	33	dmeqd 5895	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐻 ISubGr 𝑀)) = dom (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}))
35		ssrab2 4008	. . . . . . . . . . 11 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} ⊆ dom 𝐽
36	35	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} ⊆ dom 𝐽)
37		ssdmres 6017	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} ⊆ dom 𝐽 ↔ dom (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}) = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀})
38	36, 37	sylib 221	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}) = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀})
39		isubgrgrim.l	. . . . . . . . . . 11 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
40	39	eqcomi 2772	. . . . . . . . . 10 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} = 𝐿
41	40	a1i 11	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} = 𝐿)
42	34, 38, 41	3eqtrd 2802	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐻 ISubGr 𝑀)) = 𝐿)
43	18, 30, 42	f1oeq123d 6843	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ↔ 𝑔:𝐾⟶1-1onto𝐿))
44	43	anbi1d 643	. . . . . 6 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ (𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)))))
45	29	reseq2d 5982	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}) = (𝐼 ↾ 𝐾))
46	21, 45	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐺 ISubGr 𝑁)) = (𝐼 ↾ 𝐾))
47	46	fveq1d 6913	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖) = ((𝐼 ↾ 𝐾)‘𝑖))
48	47	imaeq2d 6065	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)))
49	40	reseq2i 5979	. . . . . . . . . . . . 13 ⊢ (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}) = (𝐽 ↾ 𝐿)
50	33, 49	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐻 ISubGr 𝑀)) = (𝐽 ↾ 𝐿))
51	50	fveq1d 6913	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)))
52	48, 51	eqeq12d 2779	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) ↔ (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖))))
53	30, 52	raleqbidv 3338	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ 𝐾 (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖))))
54	53	adantr 486	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) → (∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ 𝐾 (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖))))
55		fvres 6930	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝐾 → ((𝐼 ↾ 𝐾)‘𝑖) = (𝐼‘𝑖))
56	55	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑖 ∈ 𝐾) → ((𝐼 ↾ 𝐾)‘𝑖) = (𝐼‘𝑖))
57	56	imaeq2d 6065	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑖 ∈ 𝐾) → (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))
58	57	adantlr 728	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) ∧ 𝑖 ∈ 𝐾) → (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))
59		f1of 6849	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐾⟶1-1onto𝐿 → 𝑔:𝐾⟶𝐿)
60	59	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) → 𝑔:𝐾⟶𝐿)
61	60	ffvelcdmda 7111	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) ∧ 𝑖 ∈ 𝐾) → (𝑔‘𝑖) ∈ 𝐿)
62	61	fvresd 6931	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) ∧ 𝑖 ∈ 𝐾) → ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
63	58, 62	eqeq12d 2779	. . . . . . . . 9 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) ∧ 𝑖 ∈ 𝐾) → ((𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
64	63	ralbidva 3187	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) → (∀𝑖 ∈ 𝐾 (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
65	54, 64	bitrd 282	. . . . . . 7 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾⟶1-1onto𝐿) → (∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
66	65	pm5.32da 590	. . . . . 6 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ (𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
67	44, 66	bitrd 282	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ (𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
68	67	exbidv 1956	. . . 4 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ ∃𝑔(𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
69	17, 68	anbi12d 644	. . 3 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝑓:(Vtx‘(𝐺 ISubGr 𝑁))⟶1-1onto(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)))) ↔ (𝑓:𝑁⟶1-1onto𝑀 ∧ ∃𝑔(𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
70	69	exbidv 1956	. 2 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝑁))⟶1-1onto(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))⟶1-1ontodom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)))) ↔ ∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ ∃𝑔(𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
71	9, 70	bitrd 282	1 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝐺 ISubGr 𝑁) ≃_𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ ∃𝑔(𝑔:𝐾⟶1-1onto𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ∀wral 3080 {crab 3416 Vcvv 3454 ⊆ wss 3900 class class class wbr 5106 dom cdm 5656 ↾ cres 5658 “ cima 5659 ⟶wf 6557 ⟶1-1ontowf1o 6560 ‘cfv 6561 (class class class)co 7446 Vtxcvtx 29607 iEdgciedg 29608 ISubGr cisubgr 49173 ≃_𝑔𝑟 cgric 49189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-1o 8507 df-map 8880 df-vtx 29609 df-iedg 29610 df-isubgr 49174 df-grim 49191 df-gric 49194

This theorem is used by: uhgrimisgrgric 49244 clnbgrisubgrgrim 49245

W3C validator