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Theorem uspgrlimlem1 47813
Description: Lemma 1 for uspgrlim 47817. (Contributed by AV, 16-Aug-2025.)
Hypotheses
Ref Expression
uspgrlimlem1.m 𝑀 = (𝐻 ClNeighbVtx 𝑋)
uspgrlimlem1.j 𝐽 = (Edg‘𝐻)
uspgrlimlem1.l 𝐿 = {𝑥𝐽𝑥𝑀}
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.
StepHypRef Expression
1 uspgrlimlem1.l . 2 𝐿 = {𝑥𝐽𝑥𝑀}
2 eqid 2733 . . . . . 6 (iEdg‘𝐻) = (iEdg‘𝐻)
32uspgrf1oedg 29186 . . . . 5 (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
4 f1of 6843 . . . . 5 ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻))
53, 4syl 17 . . . 4 (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶(Edg‘𝐻))
6 ssrab2 4090 . . . 4 {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ⊆ dom (iEdg‘𝐻)
7 fimarab 6977 . . . 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})
85, 6, 7sylancl 585 . . 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)
109eqcomi 2742 . . . . 5 (Edg‘𝐻) = 𝐽
1110a1i 11 . . . 4 (𝐻 ∈ USPGraph → (Edg‘𝐻) = 𝐽)
12 fveq2 6901 . . . . . . 7 (𝑥 = 𝑧 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐻)‘𝑧))
1312sseq1d 4027 . . . . . 6 (𝑥 = 𝑧 → (((iEdg‘𝐻)‘𝑥) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘𝑧) ⊆ 𝑀))
1413rexrab 3705 . . . . 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 4021 . . . . . . . 8 (((iEdg‘𝐻)‘𝑧) = 𝑦 → (((iEdg‘𝐻)‘𝑧) ⊆ 𝑀𝑦𝑀))
1615biimpac 478 . . . . . . 7 ((((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) → 𝑦𝑀)
1716a1i 11 . . . . . 6 (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑧 ∈ dom (iEdg‘𝐻)) → ((((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) → 𝑦𝑀))
18 f1ocnv 6855 . . . . . . . . 9 ((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) → (iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻))
19 f1of 6843 . . . . . . . . 9 ((iEdg‘𝐻):(Edg‘𝐻)–1-1-onto→dom (iEdg‘𝐻) → (iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻))
203, 18, 193syl 18 . . . . . . . 8 (𝐻 ∈ USPGraph → (iEdg‘𝐻):(Edg‘𝐻)⟶dom (iEdg‘𝐻))
2120ffvelcdmda 7098 . . . . . . 7 ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘𝑦) ∈ dom (iEdg‘𝐻))
2221adantr 480 . . . . . 6 (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦𝑀) → ((iEdg‘𝐻)‘𝑦) ∈ dom (iEdg‘𝐻))
23 f1ocnvfv2 7290 . . . . . . . . 9 (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦)
243, 23sylan 579 . . . . . . . 8 ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦)
2524adantr 480 . . . . . . 7 (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦𝑀) → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦)
26 sseq1 4021 . . . . . . . . . . 11 (𝑦 = ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) → (𝑦𝑀 ↔ ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
2726eqcoms 2741 . . . . . . . . . 10 (((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦 → (𝑦𝑀 ↔ ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
2827biimpcd 249 . . . . . . . . 9 (𝑦𝑀 → (((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦 → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
2928adantl 481 . . . . . . . 8 (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦𝑀) → (((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦 → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
3029ancrd 551 . . . . . . 7 (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦𝑀) → (((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦 → (((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦)))
3125, 30mpd 15 . . . . . 6 (((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) ∧ 𝑦𝑀) → (((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦))
32 fveq2 6901 . . . . . . . 8 (𝑧 = ((iEdg‘𝐻)‘𝑦) → ((iEdg‘𝐻)‘𝑧) = ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)))
3332sseq1d 4027 . . . . . . 7 (𝑧 = ((iEdg‘𝐻)‘𝑦) → (((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ↔ ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) ⊆ 𝑀))
34 fveqeq2 6910 . . . . . . 7 (𝑧 = ((iEdg‘𝐻)‘𝑦) → (((iEdg‘𝐻)‘𝑧) = 𝑦 ↔ ((iEdg‘𝐻)‘((iEdg‘𝐻)‘𝑦)) = 𝑦))
3533, 34anbi12d 631 . . . . . 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)))
3617, 22, 31, 35rspceb2dv 3626 . . . . 5 ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (∃𝑧 ∈ dom (iEdg‘𝐻)(((iEdg‘𝐻)‘𝑧) ⊆ 𝑀 ∧ ((iEdg‘𝐻)‘𝑧) = 𝑦) ↔ 𝑦𝑀))
3714, 36bitrid 283 . . . 4 ((𝐻 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐻)) → (∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦𝑦𝑀))
3811, 37rabeqbidva 3449 . . 3 (𝐻 ∈ USPGraph → {𝑦 ∈ (Edg‘𝐻) ∣ ∃𝑧 ∈ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀} ((iEdg‘𝐻)‘𝑧) = 𝑦} = {𝑦𝐽𝑦𝑀})
39 sseq1 4021 . . . . 5 (𝑦 = 𝑥 → (𝑦𝑀𝑥𝑀))
4039cbvrabv 3443 . . . 4 {𝑦𝐽𝑦𝑀} = {𝑥𝐽𝑥𝑀}
4140a1i 11 . . 3 (𝐻 ∈ USPGraph → {𝑦𝐽𝑦𝑀} = {𝑥𝐽𝑥𝑀})
428, 38, 413eqtrrd 2778 . 2 (𝐻 ∈ USPGraph → {𝑥𝐽𝑥𝑀} = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
431, 42eqtrid 2785 1 (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  wrex 3066  {crab 3432  wss 3963  ccnv 5682  dom cdm 5683  cima 5686  wf 6554  1-1-ontowf1o 6557  cfv 6558  (class class class)co 7425  iEdgciedg 29010  Edgcedg 29060  USPGraphcuspgr 29161   ClNeighbVtx cclnbgr 47693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-edg 29061  df-uspgr 29163
This theorem is referenced by:  uspgrlim  47817
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