Theorem uspgrlimlem1 48571

Description: Lemma 1 for uspgrlim 48575. (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 2761	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
3	2	uspgrf1oedg 29331	. . . . 5 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
4		f1of 6801	. . . . 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 4031	. . . 4 ⊢ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ⊆ dom (iEdg‘𝐻)
7		fimarab 6936	. . . 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 595	. . 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 2770	. . . . 5 ⊢ (Edg‘𝐻) = 𝐽
11	10	a1i 11	. . . 4 ⊢ (𝐻 ∈ USPGraph → (Edg‘𝐻) = 𝐽)
12		fveq2 6862	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐻)‘𝑧))
13	12	sseq1d 3965	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((iEdg‘𝐻)‘𝑥) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘𝑧) ⊆ 𝑀))
14	13	rexrab 3657	. . . . 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 3959	. . . . . . . 8 ⊢ (((iEdg‘𝐻)‘𝑧) = 𝑦 → (((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ↔ 𝑦 ⊆ 𝑀))
16	15	biimpac 482	. . . . . . 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 6814	. . . . . . . . 9 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → ◡(iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻))
19		f1of 6801	. . . . . . . . 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 7060	. . . . . . 7 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (◡(iEdg‘𝐻)‘𝑦) ∈ dom (iEdg‘𝐻))
22	21	adantr 484	. . . . . 6 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → (◡(iEdg‘𝐻)‘𝑦) ∈ dom (iEdg‘𝐻))
23		f1ocnvfv2 7256	. . . . . . . . 9 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
24	3, 23	sylan 589	. . . . . . . 8 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
25	24	adantr 484	. . . . . . 7 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦)
26		sseq1 3959	. . . . . . . . . . 11 ⊢ (𝑦 = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) → (𝑦 ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
27	26	eqcoms 2769	. . . . . . . . . 10 ⊢ (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → (𝑦 ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
28	27	biimpcd 251	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑀 → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
29	28	adantl 485	. . . . . . . 8 ⊢ (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦 ⊆ 𝑀) → (((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦 → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
30	29	ancrd 559	. . . . . . 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 6862	. . . . . . . 8 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → ((iEdg‘𝐻)‘𝑧) = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)))
33	32	sseq1d 3965	. . . . . . 7 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → (((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
34		fveqeq2 6871	. . . . . . 7 ⊢ (𝑧 = (◡(iEdg‘𝐻)‘𝑦) → (((iEdg‘𝐻)‘𝑧) = 𝑦 ↔ ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘𝑦)) = 𝑦))
35	33, 34	anbi12d 641	. . . . . 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 3584	. . . . 5 ⊢ ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (∃𝑧 ∈ dom (iEdg‘𝐻)(((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) ↔ 𝑦 ⊆ 𝑀))
37	14, 36	bitrid 285	. . . 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 3429	. . 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 3959	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑀 ↔ 𝑥 ⊆ 𝑀))
40	39	cbvrabv 3423	. . . 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 2801	. 2 ⊢ (𝐻 ∈ USPGraph → {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀} = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
43	1, 42	eqtrid 2808	1 ⊢ (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413   ⊆ wss 3902  ^◡ccnv 5642  dom cdm 5643   “ cima 5646  ⟶wf 6512  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391  iEdgciedg 29155  Edgcedg 29205  USPGraphcuspgr 29306   ClNeighbVtx cclnbgr 48401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-edg 29206  df-uspgr 29308

This theorem is referenced by:  uspgrlim  48575