Theorem uspgrlimlem2 47978

Description: Lemma 2 for uspgrlim 47981. (Contributed by AV, 16-Aug-2025.)

Hypotheses

Ref	Expression
uspgrlimlem1.m	⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑋)
uspgrlimlem1.j	⊢ 𝐽 = (Edg‘𝐻)
uspgrlimlem1.l	⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}

Assertion

Ref	Expression
uspgrlimlem2	⊢ (𝐻 ∈ USPGraph → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})

Distinct variable groups:   𝑥,𝐻   𝑥,𝐽   𝑥,𝑀   𝑥,𝐿

Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem uspgrlimlem2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
2	1	uspgrf1oedg 29106	. . . 4 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
3		f1ocnv 6814	. . . 4 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → ◡(iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻))
4		f1of 6802	. . . 4 ⊢ (◡(iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻) → ◡(iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻))
5	2, 3, 4	3syl 18	. . 3 ⊢ (𝐻 ∈ USPGraph → ◡(iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻))
6		uspgrlimlem1.l	. . . . 5 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
7		uspgrlimlem1.j	. . . . . 6 ⊢ 𝐽 = (Edg‘𝐻)
8	7	rabeqi 3422	. . . . 5 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀} = {𝑥 ∈ (Edg‘𝐻) ∣ 𝑥 ⊆ 𝑀}
9	6, 8	eqtri 2753	. . . 4 ⊢ 𝐿 = {𝑥 ∈ (Edg‘𝐻) ∣ 𝑥 ⊆ 𝑀}
10	9	ssrab3 4047	. . 3 ⊢ 𝐿 ⊆ (Edg‘𝐻)
11		fimarab 6937	. . 3 ⊢ ((◡(iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻) ∧ 𝐿 ⊆ (Edg‘𝐻)) → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥})
12	5, 10, 11	sylancl 586	. 2 ⊢ (𝐻 ∈ USPGraph → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥})
13		sseq1 3974	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑀 ↔ 𝑦 ⊆ 𝑀))
14	13, 6	elrab2 3664	. . . . . 6 ⊢ (𝑦 ∈ 𝐿 ↔ (𝑦 ∈ 𝐽 ∧ 𝑦 ⊆ 𝑀))
15	7	eleq2i 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐽 ↔ 𝑦 ∈ (Edg‘𝐻))
16	15	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ∈ (Edg‘𝐻))
17		f1ocnvfv2 7254	. . . . . . . . . . . . . . . 16 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
18	2, 16, 17	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ 𝐽) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
19	18	eqcomd 2736	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ 𝐽) → 𝑦 = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)))
20	19	sseq1d 3980	. . . . . . . . . . . . 13 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ 𝐽) → (𝑦 ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
21	20	biimpd 229	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ 𝐽) → (𝑦 ⊆ 𝑀 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
22	21	ex 412	. . . . . . . . . . 11 ⊢ (𝐻 ∈ USPGraph → (𝑦 ∈ 𝐽 → (𝑦 ⊆ 𝑀 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀)))
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (𝑦 ∈ 𝐽 → (𝑦 ⊆ 𝑀 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀)))
24	23	imp32 418	. . . . . . . . 9 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑦 ⊆ 𝑀)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀)
25	24	3adant3 1132	. . . . . . . 8 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑦 ⊆ 𝑀) ∧ (◡(iEdg‘𝐻)‘𝑦) = 𝑥) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀)
26		fveq2 6860	. . . . . . . . . 10 ⊢ ((◡(iEdg‘𝐻)‘𝑦) = 𝑥 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = ((iEdg‘𝐻)‘𝑥))
27	26	sseq1d 3980	. . . . . . . . 9 ⊢ ((◡(iEdg‘𝐻)‘𝑦) = 𝑥 → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
28	27	3ad2ant3 1135	. . . . . . . 8 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑦 ⊆ 𝑀) ∧ (◡(iEdg‘𝐻)‘𝑦) = 𝑥) → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
29	25, 28	mpbid 232	. . . . . . 7 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑦 ⊆ 𝑀) ∧ (◡(iEdg‘𝐻)‘𝑦) = 𝑥) → ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀)
30	29	3exp 1119	. . . . . 6 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → ((𝑦 ∈ 𝐽 ∧ 𝑦 ⊆ 𝑀) → ((◡(iEdg‘𝐻)‘𝑦) = 𝑥 → ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀)))
31	14, 30	biimtrid 242	. . . . 5 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (𝑦 ∈ 𝐿 → ((◡(iEdg‘𝐻)‘𝑦) = 𝑥 → ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀)))
32	31	rexlimdv 3133	. . . 4 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥 → ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
33		fveqeq2 6869	. . . . . 6 ⊢ (𝑦 = ((iEdg‘𝐻)‘𝑥) → ((◡(iEdg‘𝐻)‘𝑦) = 𝑥 ↔ (◡(iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑥)) = 𝑥))
34		f1of 6802	. . . . . . . . . 10 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻))
35		eqid 2730	. . . . . . . . . . . 12 ⊢ dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
36	7	eqcomi 2739	. . . . . . . . . . . 12 ⊢ (Edg‘𝐻) = 𝐽
37	35, 36	feq23i 6684	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻) ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)⟶𝐽)
38	37	biimpi 216	. . . . . . . . . 10 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶𝐽)
39	2, 34, 38	3syl 18	. . . . . . . . 9 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶𝐽)
40	39	ffvelcdmda 7058	. . . . . . . 8 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘𝑥) ∈ 𝐽)
41	40	anim1i 615	. . . . . . 7 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀) → (((iEdg‘𝐻)‘𝑥) ∈ 𝐽 ∧ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
42		sseq1 3974	. . . . . . . 8 ⊢ (𝑦 = ((iEdg‘𝐻)‘𝑥) → (𝑦 ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
43	13, 42, 6	elrab2w 3665	. . . . . . 7 ⊢ (((iEdg‘𝐻)‘𝑥) ∈ 𝐿 ↔ (((iEdg‘𝐻)‘𝑥) ∈ 𝐽 ∧ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
44	41, 43	sylibr 234	. . . . . 6 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀) → ((iEdg‘𝐻)‘𝑥) ∈ 𝐿)
45		f1ocnvfv1 7253	. . . . . . . 8 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (◡(iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑥)) = 𝑥)
46	2, 45	sylan 580	. . . . . . 7 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (◡(iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑥)) = 𝑥)
47	46	adantr 480	. . . . . 6 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀) → (◡(iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑥)) = 𝑥)
48	33, 44, 47	rspcedvdw 3594	. . . . 5 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀) → ∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥)
49	48	ex 412	. . . 4 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘𝑥) ⊆ 𝑀 → ∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥))
50	32, 49	impbid 212	. . 3 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑥 ∈ dom (iEdg‘𝐻)) → (∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥 ↔ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀))
51	50	rabbidva 3415	. 2 ⊢ (𝐻 ∈ USPGraph → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ∃𝑦 ∈ 𝐿 (◡(iEdg‘𝐻)‘𝑦) = 𝑥} = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})
52	12, 51	eqtrd 2765	1 ⊢ (𝐻 ∈ USPGraph → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  {crab 3408   ⊆ wss 3916  ^◡ccnv 5639  dom cdm 5640   “ cima 5643  ⟶wf 6509  –1-1-onto→wf1o 6512  ‘cfv 6513  (class class class)co 7389  iEdgciedg 28930  Edgcedg 28980  USPGraphcuspgr 29081   ClNeighbVtx cclnbgr 47809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-edg 28981  df-uspgr 29083

This theorem is referenced by:  uspgrlim  47981