isubgrgrim - Mathbox for Alexander van der Vekens

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Theorem isubgrgrim 48971

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–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀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 7445	. . . 4 ⊢ (𝐺 ISubGr 𝑁) ∈ V
2		ovex 7445	. . . 4 ⊢ (𝐻 ISubGr 𝑀) ∈ V
3	1, 2	pm3.2i 476	. . 3 ⊢ ((𝐺 ISubGr 𝑁) ∈ V ∧ (𝐻 ISubGr 𝑀) ∈ V)
4		eqid 2761	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑁)) = (Vtx‘(𝐺 ISubGr 𝑁))
5		eqid 2761	. . . 4 ⊢ (Vtx‘(𝐻 ISubGr 𝑀)) = (Vtx‘(𝐻 ISubGr 𝑀))
6		eqid 2761	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑁)) = (iEdg‘(𝐺 ISubGr 𝑁))
7		eqid 2761	. . . 4 ⊢ (iEdg‘(𝐻 ISubGr 𝑀)) = (iEdg‘(𝐻 ISubGr 𝑀))
8	4, 5, 6, 7	dfgric2 48957	. . 3 ⊢ (((𝐺 ISubGr 𝑁) ∈ V ∧ (𝐻 ISubGr 𝑀) ∈ V) → ((𝐺 ISubGr 𝑁) ≃_𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝑁))–1-1-onto→(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (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))–1-1-onto→(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))))))
10		eqidd 2762	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → 𝑓 = 𝑓)
11		isubgrgrim.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	11	isubgrvtx 48909	. . . . . 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 48909	. . . . . 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 6810	. . . 4 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑓:(Vtx‘(𝐺 ISubGr 𝑁))–1-1-onto→(Vtx‘(𝐻 ISubGr 𝑀)) ↔ 𝑓:𝑁–1-1-onto→𝑀))
18		eqidd 2762	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → 𝑔 = 𝑔)
19		isubgrgrim.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (iEdg‘𝐺)
20	11, 19	isubgriedg 48905	. . . . . . . . . . 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 5887	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐺 ISubGr 𝑁)) = dom (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}))
23		ssrab2 4028	. . . . . . . . . . 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 6004	. . . . . . . . . 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 2770	. . . . . . . . . 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 2800	. . . . . . . 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 48905	. . . . . . . . . . 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 5887	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐻 ISubGr 𝑀)) = dom (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}))
35		ssrab2 4028	. . . . . . . . . . 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 6004	. . . . . . . . . 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 2770	. . . . . . . . . 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 2800	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐻 ISubGr 𝑀)) = 𝐿)
43	18, 30, 42	f1oeq123d 6810	. . . . . . 7 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (iEdg‘(𝐻 ISubGr 𝑀)) ↔ 𝑔:𝐾–1-1-onto→𝐿))
44	43	anbi1d 643	. . . . . 6 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)))))
45	29	reseq2d 5970	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}) = (𝐼 ↾ 𝐾))
46	21, 45	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐺 ISubGr 𝑁)) = (𝐼 ↾ 𝐾))
47	46	fveq1d 6879	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖) = ((𝐼 ↾ 𝐾)‘𝑖))
48	47	imaeq2d 6054	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)))
49	40	reseq2i 5967	. . . . . . . . . . . . 13 ⊢ (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}) = (𝐽 ↾ 𝐿)
50	33, 49	eqtrdi 2812	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐻 ISubGr 𝑀)) = (𝐽 ↾ 𝐿))
51	50	fveq1d 6879	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)))
52	48, 51	eqeq12d 2777	. . . . . . . . . 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 3335	. . . . . . . . 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–1-1-onto→𝐿) → (∀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 6896	. . . . . . . . . . . . 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 6054	. . . . . . . . . . 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–1-1-onto→𝐿) ∧ 𝑖 ∈ 𝐾) → (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))
59		f1of 6816	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐾–1-1-onto→𝐿 → 𝑔:𝐾⟶𝐿)
60	59	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾–1-1-onto→𝐿) → 𝑔:𝐾⟶𝐿)
61	60	ffvelcdmda 7076	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ 𝑖 ∈ 𝐾) → (𝑔‘𝑖) ∈ 𝐿)
62	61	fvresd 6897	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ 𝑖 ∈ 𝐾) → ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
63	58, 62	eqeq12d 2777	. . . . . . . . 9 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ 𝑖 ∈ 𝐾) → ((𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
64	63	ralbidva 3184	. . . . . . . 8 ⊢ ((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾–1-1-onto→𝐿) → (∀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–1-1-onto→𝐿) → (∀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–1-1-onto→𝐿 ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀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))–1-1-onto→dom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖))) ↔ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
68	67	exbidv 1954	. . . 4 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (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–1-1-onto→𝐿 ∧ ∀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))–1-1-onto→(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (iEdg‘(𝐻 ISubGr 𝑀)) ∧ ∀𝑖 ∈ dom (iEdg‘(𝐺 ISubGr 𝑁))(𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)))) ↔ (𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
70	69	exbidv 1954	. 2 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝑁))–1-1-onto→(Vtx‘(𝐻 ISubGr 𝑀)) ∧ ∃𝑔(𝑔:dom (iEdg‘(𝐺 ISubGr 𝑁))–1-1-onto→dom (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–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀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–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀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 2145 ∀wral 3077 {crab 3413 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 dom cdm 5651 ↾ cres 5653 “ cima 5654 ⟶wf 6527 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7412 Vtxcvtx 29556 iEdgciedg 29557 ISubGr cisubgr 48902 ≃_𝑔𝑟 cgric 48918

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-1o 8460 df-map 8833 df-vtx 29558 df-iedg 29559 df-isubgr 48903 df-grim 48920 df-gric 48923

This theorem is used by: uhgrimisgrgric 48973 clnbgrisubgrgrim 48974

W3C validator