isubgrgrim - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgrgrim		Structured version Visualization version GIF version

Theorem isubgrgrim 49120

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 7453	. . . 4 ⊢ (𝐺 ISubGr 𝑁) ∈ V
2		ovex 7453	. . . 4 ⊢ (𝐻 ISubGr 𝑀) ∈ V
3	1, 2	pm3.2i 476	. . 3 ⊢ ((𝐺 ISubGr 𝑁) ∈ V ∧ (𝐻 ISubGr 𝑀) ∈ V)
4		eqid 2760	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑁)) = (Vtx‘(𝐺 ISubGr 𝑁))
5		eqid 2760	. . . 4 ⊢ (Vtx‘(𝐻 ISubGr 𝑀)) = (Vtx‘(𝐻 ISubGr 𝑀))
6		eqid 2760	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑁)) = (iEdg‘(𝐺 ISubGr 𝑁))
7		eqid 2760	. . . 4 ⊢ (iEdg‘(𝐻 ISubGr 𝑀)) = (iEdg‘(𝐻 ISubGr 𝑀))
8	4, 5, 6, 7	dfgric2 49106	. . 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 2761	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → 𝑓 = 𝑓)
11		isubgrgrim.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	11	isubgrvtx 49058	. . . . . 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 49058	. . . . . 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 6818	. . . 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 2761	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → 𝑔 = 𝑔)
19		isubgrgrim.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (iEdg‘𝐺)
20	11, 19	isubgriedg 49054	. . . . . . . . . . 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 5886	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐺 ISubGr 𝑁)) = dom (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}))
23		ssrab2 4005	. . . . . . . . . . 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 6003	. . . . . . . . . 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 2769	. . . . . . . . . 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 2799	. . . . . . . 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 49054	. . . . . . . . . . 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 5886	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐻 ISubGr 𝑀)) = dom (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}))
35		ssrab2 4005	. . . . . . . . . . 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 6003	. . . . . . . . . 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 2769	. . . . . . . . . 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 2799	. . . . . . . 8 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → dom (iEdg‘(𝐻 ISubGr 𝑀)) = 𝐿)
43	18, 30, 42	f1oeq123d 6818	. . . . . . 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 5969	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝐼 ↾ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}) = (𝐼 ↾ 𝐾))
46	21, 45	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐺 ISubGr 𝑁)) = (𝐼 ↾ 𝐾))
47	46	fveq1d 6887	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖) = ((𝐼 ↾ 𝐾)‘𝑖))
48	47	imaeq2d 6051	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (𝑓 “ ((iEdg‘(𝐺 ISubGr 𝑁))‘𝑖)) = (𝑓 “ ((𝐼 ↾ 𝐾)‘𝑖)))
49	40	reseq2i 5966	. . . . . . . . . . . . 13 ⊢ (𝐽 ↾ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}) = (𝐽 ↾ 𝐿)
50	33, 49	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → (iEdg‘(𝐻 ISubGr 𝑀)) = (𝐽 ↾ 𝐿))
51	50	fveq1d 6887	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((iEdg‘(𝐻 ISubGr 𝑀))‘(𝑔‘𝑖)) = ((𝐽 ↾ 𝐿)‘(𝑔‘𝑖)))
52	48, 51	eqeq12d 2776	. . . . . . . . . 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 3334	. . . . . . . . 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 6904	. . . . . . . . . . . . 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 6051	. . . . . . . . . . 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 6824	. . . . . . . . . . . . 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 7084	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ 𝑖 ∈ 𝐾) → (𝑔‘𝑖) ∈ 𝐿)
62	61	fvresd 6905	. . . . . . . . . 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 2776	. . . . . . . . 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 3183	. . . . . . . 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 1953	. . . 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 1953	. 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 1569 ∃wex 1811 ∈ wcel 2144 ∀wral 3076 {crab 3412 Vcvv 3450 ⊆ wss 3895 class class class wbr 5102 dom cdm 5650 ↾ cres 5652 “ cima 5653 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 Vtxcvtx 29579 iEdgciedg 29580 ISubGr cisubgr 49051 ≃_𝑔𝑟 cgric 49067

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-1o 8479 df-map 8852 df-vtx 29581 df-iedg 29582 df-isubgr 49052 df-grim 49069 df-gric 49072

This theorem is used by: uhgrimisgrgric 49122 clnbgrisubgrgrim 49123

W3C validator