Theorem uspgrlimlem1 48019

Description: Lemma 1 for uspgrlim 48023. (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
uspgrlimlem1	⊢ (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))

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

Allowed substitution hints:   𝐿(𝑥)   𝑋(𝑥)

Proof of Theorem uspgrlimlem1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrlimlem1.l	. 2 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
2		eqid 2731	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
3	2	uspgrf1oedg 29146	. . . . 5 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
4		f1of 6758	. . . . 5 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻))
5	3, 4	syl 17	. . . 4 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻))
6		ssrab2 4025	. . . 4 ⊢ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ⊆ dom (iEdg‘𝐻)
7		fimarab 6891	. . . 4 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻) ∧ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ⊆ dom (iEdg‘𝐻)) → ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}) = {𝑦 ∈ (Edg‘𝐻) ∣ ∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦})
8	5, 6, 7	sylancl 586	. . 3 ⊢ (𝐻 ∈ USPGraph → ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}) = {𝑦 ∈ (Edg‘𝐻) ∣ ∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦})
9		uspgrlimlem1.j	. . . . . 6 ⊢ 𝐽 = (Edg‘𝐻)
10	9	eqcomi 2740	. . . . 5 ⊢ (Edg‘𝐻) = 𝐽
11	10	a1i 11	. . . 4 ⊢ (𝐻 ∈ USPGraph → (Edg‘𝐻) = 𝐽)
12		fveq2 6817	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐻)‘𝑧))
13	12	sseq1d 3961	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((iEdg‘𝐻)‘𝑥) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘𝑧) ⊆ 𝑀))
14	13	rexrab 3650	. . . . 5 ⊢ (∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ dom (iEdg‘𝐻)(((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦))
15		sseq1 3955	. . . . . . . 8 ⊢ (((iEdg‘𝐻)‘𝑧) = 𝑦 → (((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ↔ 𝑦 ⊆ 𝑀))
16	15	biimpac 478	. . . . . . 7 ⊢ ((((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) → 𝑦 ⊆ 𝑀)
17	16	a1i 11	. . . . . 6 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑧 ∈ dom (iEdg‘𝐻)) → ((((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) → 𝑦 ⊆ 𝑀))
18		f1ocnv 6770	. . . . . . . . 9 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → ◡(iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻))
19		f1of 6758	. . . . . . . . 9 ⊢ (◡(iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻) → ◡(iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻))
20	3, 18, 19	3syl 18	. . . . . . . 8 ⊢ (𝐻 ∈ USPGraph → ◡(iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻))
21	20	ffvelcdmda 7012	. . . . . . 7 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (◡(iEdg‘𝐻)‘𝑦) ∈ dom (iEdg‘𝐻))
22	21	adantr 480	. . . . . 6 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → (◡(iEdg‘𝐻)‘𝑦) ∈ dom (iEdg‘𝐻))
23		f1ocnvfv2 7206	. . . . . . . . 9 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
24	3, 23	sylan 580	. . . . . . . 8 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
25	24	adantr 480	. . . . . . 7 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
26		sseq1 3955	. . . . . . . . . . 11 ⊢ (𝑦 = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) → (𝑦 ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
27	26	eqcoms 2739	. . . . . . . . . 10 ⊢ (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → (𝑦 ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
28	27	biimpcd 249	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑀 → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
29	28	adantl 481	. . . . . . . 8 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
30	29	ancrd 551	. . . . . . 7 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)))
31	25, 30	mpd 15	. . . . . 6 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦))
32		fveq2 6817	. . . . . . . 8 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → ((iEdg‘𝐻)‘𝑧) = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)))
33	32	sseq1d 3961	. . . . . . 7 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → (((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
34		fveqeq2 6826	. . . . . . 7 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → (((iEdg‘𝐻)‘𝑧) = 𝑦 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦))
35	33, 34	anbi12d 632	. . . . . 6 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → ((((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) ↔ (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)))
36	17, 22, 31, 35	rspceb2dv 3576	. . . . 5 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (∃𝑧 ∈ dom (iEdg‘𝐻)(((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) ↔ 𝑦 ⊆ 𝑀))
37	14, 36	bitrid 283	. . . 4 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦 ↔ 𝑦 ⊆ 𝑀))
38	11, 37	rabeqbidva 3411	. . 3 ⊢ (𝐻 ∈ USPGraph → {𝑦 ∈ (Edg‘𝐻) ∣ ∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑦 ⊆ 𝑀})
39		sseq1 3955	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑀 ↔ 𝑥 ⊆ 𝑀))
40	39	cbvrabv 3405	. . . 4 ⊢ {𝑦 ∈ 𝐽 ∣ 𝑦 ⊆ 𝑀} = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
41	40	a1i 11	. . 3 ⊢ (𝐻 ∈ USPGraph → {𝑦 ∈ 𝐽 ∣ 𝑦 ⊆ 𝑀} = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀})
42	8, 38, 41	3eqtrrd 2771	. 2 ⊢ (𝐻 ∈ USPGraph → {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀} = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
43	1, 42	eqtrid 2778	1 ⊢ (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395   ⊆ wss 3897  ^◡ccnv 5610  dom cdm 5611   “ cima 5614  ⟶wf 6472  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341  iEdgciedg 28970  Edgcedg 29020  USPGraphcuspgr 29121   ClNeighbVtx cclnbgr 47849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-edg 29021  df-uspgr 29123

This theorem is referenced by:  uspgrlim  48023