Theorem grlimpredg 48584

Description: For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge in 𝐻. (Contributed by AV, 27-Dec-2025.)

Hypotheses

Ref	Expression
clnbgrvtxedg.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
clnbgrvtxedg.i	⊢ 𝐼 = (Edg‘𝐺)
clnbgrvtxedg.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
grlimedgclnbgr.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))
grlimedgclnbgr.j	⊢ 𝐽 = (Edg‘𝐻)
grlimedgclnbgr.l	⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}

Assertion

Ref	Expression
grlimpredg	⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))

Distinct variable groups:   𝑥,𝐼   𝑥,𝑁   𝑥,𝐴,𝑓   𝑓,𝐹,𝑥   𝑓,𝐺,𝑥   𝑓,𝐻,𝑥   𝑓,𝐼   𝑥,𝐽   𝐵,𝑓,𝑥   𝑓,𝑉   𝑓,𝑊   𝑥,𝑀   𝑥,𝑓

Allowed substitution hints:   𝐽(𝑓)   𝐾(𝑥,𝑓)   𝐿(𝑥,𝑓)   𝑀(𝑓)   𝑁(𝑓)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem grlimpredg

Step	Hyp	Ref	Expression
1		clnbgrvtxedg.n	. . 3 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
2		clnbgrvtxedg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐺)
3		clnbgrvtxedg.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
4		grlimedgclnbgr.m	. . 3 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))
5		grlimedgclnbgr.j	. . 3 ⊢ 𝐽 = (Edg‘𝐻)
6		grlimedgclnbgr.l	. . 3 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
7	1, 2, 3, 4, 5, 6	grlimprclnbgredg 48583	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
8		sseq1 3961	. . . . . 6 ⊢ (𝑥 = {(𝑓‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
9	8, 6	elrab2 3653	. . . . 5 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
10		simpl 486	. . . . . 6 ⊢ (({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀) → {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)
11	10	a1i 11	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → (({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀) → {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))
12	9, 11	biimtrid 244	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 → {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))
13	12	imdistanda 579	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ((𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)))
14	13	eximdv 1936	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)))
15	7, 14	mpd 15	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {crab 3413   ⊆ wss 3904  {cpr 4583  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  Edgcedg 29194  USPGraphcuspgr 29295   ClNeighbVtx cclnbgr 48404   GraphLocIso cgrlim 48562

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 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  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-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-n0 12479  df-xnn0 12552  df-z 12566  df-uz 12837  df-fz 13510  df-hash 14341  df-vtx 29145  df-iedg 29146  df-edg 29195  df-uhgr 29205  df-upgr 29229  df-uspgr 29297  df-nbgr 29480  df-clnbgr 48405  df-isubgr 48447  df-grim 48464  df-gric 48467  df-grlim 48564

This theorem is referenced by: (None)