Theorem grlimgrtrilem2 49208

Description: Lemma 3 for grlimgrtri 49209. (Contributed by AV, 23-Aug-2025.)

Hypotheses

Ref	Expression
grlimgrtrilem1.v	⊢ 𝑉 = (Vtx‘𝐺)
grlimgrtrilem1.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)
grlimgrtrilem1.i	⊢ 𝐼 = (Edg‘𝐺)
grlimgrtrilem1.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
grlimgrtrilem2.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑎))
grlimgrtrilem2.j	⊢ 𝐽 = (Edg‘𝐻)
grlimgrtrilem2.l	⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}

Assertion

Ref	Expression
grlimgrtrilem2	⊢ (((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) ∧ {𝑏, 𝑐} ∈ 𝐾) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)

Distinct variable groups:   𝑥,𝐼   𝑥,𝑁   𝑥,𝑎   𝑥,𝑏   𝑥,𝑐   𝑥,𝐽   𝑖,𝐾   𝑖,𝑏   𝑖,𝑐   𝑓,𝑖   𝑔,𝑖

Allowed substitution hints:   𝐹(𝑥, 𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝐺(𝑥, 𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝐻(𝑥, 𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝐼(𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝐽(𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝐾(𝑥, 𝑓, 𝑔, 𝑎, 𝑏, 𝑐)   𝐿(𝑥, 𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝑀(𝑥, 𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝑁(𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)   𝑉(𝑥, 𝑓, 𝑔, 𝑖, 𝑎, 𝑏, 𝑐)

Proof of Theorem grlimgrtrilem2

Step	Hyp	Ref	Expression
1		imaeq2 6050	. . . . 5 ⊢ (𝑖 = {𝑏, 𝑐} → (𝑓 “ 𝑖) = (𝑓 “ {𝑏, 𝑐}))
2		fveq2 6890	. . . . 5 ⊢ (𝑖 = {𝑏, 𝑐} → (𝑔‘𝑖) = (𝑔‘{𝑏, 𝑐}))
3	1, 2	eqeq12d 2776	. . . 4 ⊢ (𝑖 = {𝑏, 𝑐} → ((𝑓 “ 𝑖) = (𝑔‘𝑖) ↔ (𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐})))
4	3	rspcv 3572	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → (∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) → (𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐})))
5		f1ofn 6830	. . . . . . . . . 10 ⊢ (𝑓:𝑁–1-1-onto→𝑀 → 𝑓 Fn 𝑁)
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → 𝑓 Fn 𝑁)
7	6	adantl 487	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑓 Fn 𝑁)
8		grlimgrtrilem1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
9	8	eleq2i 2852	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ {𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
10		sseq1 3934	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
11	10	elrab 3644	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁} ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
12	9, 11	bitri 278	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
13		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
14		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
15	13, 14	prss 4775	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) ↔ {𝑏, 𝑐} ⊆ 𝑁)
16		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁)
17	15, 16	sylbir 238	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ⊆ 𝑁 → 𝑏 ∈ 𝑁)
18	12, 17	simplbiim 514	. . . . . . . . 9 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → 𝑏 ∈ 𝑁)
19	18	adantr 486	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑏 ∈ 𝑁)
20		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
21	15, 20	sylbir 238	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ⊆ 𝑁 → 𝑐 ∈ 𝑁)
22	12, 21	simplbiim 514	. . . . . . . . 9 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → 𝑐 ∈ 𝑁)
23	22	adantr 486	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑐 ∈ 𝑁)
24		fnimapr 6973	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑁 ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑓 “ {𝑏, 𝑐}) = {(𝑓‘𝑏), (𝑓‘𝑐)})
25	7, 19, 23, 24	syl3anc 1398	. . . . . . 7 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → (𝑓 “ {𝑏, 𝑐}) = {(𝑓‘𝑏), (𝑓‘𝑐)})
26	25	eqeq1d 2762	. . . . . 6 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → ((𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐}) ↔ {(𝑓‘𝑏), (𝑓‘𝑐)} = (𝑔‘{𝑏, 𝑐})))
27		grlimgrtrilem2.l	. . . . . . . . 9 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
28		ssrab2 4004	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀} ⊆ 𝐽
29	27, 28	eqsstri 3955	. . . . . . . 8 ⊢ 𝐿 ⊆ 𝐽
30		f1of 6829	. . . . . . . . . . 11 ⊢ (𝑔:𝐾–1-1-onto→𝐿 → 𝑔:𝐾⟶𝐿)
31	30	adantl 487	. . . . . . . . . 10 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → 𝑔:𝐾⟶𝐿)
32	31	adantl 487	. . . . . . . . 9 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑔:𝐾⟶𝐿)
33		simpl 488	. . . . . . . . 9 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → {𝑏, 𝑐} ∈ 𝐾)
34	32, 33	ffvelcdmd 7090	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → (𝑔‘{𝑏, 𝑐}) ∈ 𝐿)
35	29, 34	sselid 3907	. . . . . . 7 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → (𝑔‘{𝑏, 𝑐}) ∈ 𝐽)
36		eleq1 2848	. . . . . . 7 ⊢ ({(𝑓‘𝑏), (𝑓‘𝑐)} = (𝑔‘{𝑏, 𝑐}) → ({(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽 ↔ (𝑔‘{𝑏, 𝑐}) ∈ 𝐽))
37	35, 36	syl5ibrcom 250	. . . . . 6 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → ({(𝑓‘𝑏), (𝑓‘𝑐)} = (𝑔‘{𝑏, 𝑐}) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽))
38	26, 37	sylbid 243	. . . . 5 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → ((𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐}) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽))
39	38	ex 418	. . . 4 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → ((𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐}) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)))
40	39	com23 87	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → ((𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐}) → ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)))
41	4, 40	syld 48	. 2 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → (∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) → ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)))
42	41	3imp31 1129	1 ⊢ (((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) ∧ {𝑏, 𝑐} ∈ 𝐾) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3896  {cpr 4586   “ cima 5654   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7424  Vtxcvtx 29583  Edgcedg 29634   ClNeighbVtx cclnbgr 49024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-f1o 6550  df-fv 6551

This theorem is used by:  grlimgrtri  49209