Theorem grlimgrtrilem2 48493

Description: Lemma 3 for grlimgrtri 48494. (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 6008	. . . . 5 ⊢ (𝑖 = {𝑏, 𝑐} → (𝑓 “ 𝑖) = (𝑓 “ {𝑏, 𝑐}))
2		fveq2 6827	. . . . 5 ⊢ (𝑖 = {𝑏, 𝑐} → (𝑔‘𝑖) = (𝑔‘{𝑏, 𝑐}))
3	1, 2	eqeq12d 2755	. . . 4 ⊢ (𝑖 = {𝑏, 𝑐} → ((𝑓 “ 𝑖) = (𝑔‘𝑖) ↔ (𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐})))
4	3	rspcv 3556	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → (∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) → (𝑓 “ {𝑏, 𝑐}) = (𝑔‘{𝑏, 𝑐})))
5		f1ofn 6768	. . . . . . . . . 10 ⊢ (𝑓:𝑁–1-1-onto→𝑀 → 𝑓 Fn 𝑁)
6	5	adantr 481	. . . . . . . . 9 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → 𝑓 Fn 𝑁)
7	6	adantl 482	. . . . . . . 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 2831	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ {𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
10		sseq1 3940	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
11	10	elrab 3629	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁} ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
12	9, 11	bitri 276	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
13		vex 3435	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
14		vex 3435	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
15	13, 14	prss 4751	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) ↔ {𝑏, 𝑐} ⊆ 𝑁)
16		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁)
17	15, 16	sylbir 236	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ⊆ 𝑁 → 𝑏 ∈ 𝑁)
18	12, 17	simplbiim 509	. . . . . . . . 9 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → 𝑏 ∈ 𝑁)
19	18	adantr 481	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑏 ∈ 𝑁)
20		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
21	15, 20	sylbir 236	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ⊆ 𝑁 → 𝑐 ∈ 𝑁)
22	12, 21	simplbiim 509	. . . . . . . . 9 ⊢ ({𝑏, 𝑐} ∈ 𝐾 → 𝑐 ∈ 𝑁)
23	22	adantr 481	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑐 ∈ 𝑁)
24		fnimapr 6910	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑁 ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑓 “ {𝑏, 𝑐}) = {(𝑓‘𝑏), (𝑓‘𝑐)})
25	7, 19, 23, 24	syl3anc 1379	. . . . . . 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 2741	. . . . . 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 4011	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀} ⊆ 𝐽
29	27, 28	eqsstri 3961	. . . . . . . 8 ⊢ 𝐿 ⊆ 𝐽
30		f1of 6767	. . . . . . . . . . 11 ⊢ (𝑔:𝐾–1-1-onto→𝐿 → 𝑔:𝐾⟶𝐿)
31	30	adantl 482	. . . . . . . . . 10 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) → 𝑔:𝐾⟶𝐿)
32	31	adantl 482	. . . . . . . . 9 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → 𝑔:𝐾⟶𝐿)
33		simpl 483	. . . . . . . . 9 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → {𝑏, 𝑐} ∈ 𝐾)
34	32, 33	ffvelcdmd 7026	. . . . . . . 8 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → (𝑔‘{𝑏, 𝑐}) ∈ 𝐿)
35	29, 34	sselid 3913	. . . . . . 7 ⊢ (({𝑏, 𝑐} ∈ 𝐾 ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿)) → (𝑔‘{𝑏, 𝑐}) ∈ 𝐽)
36		eleq1 2827	. . . . . . 7 ⊢ ({(𝑓‘𝑏), (𝑓‘𝑐)} = (𝑔‘{𝑏, 𝑐}) → ({(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽 ↔ (𝑔‘{𝑏, 𝑐}) ∈ 𝐽))
37	35, 36	syl5ibrcom 248	. . . . . 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 241	. . . . 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 413	. . . 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 86	. . 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 47	. 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 1117	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)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391   ⊆ wss 3883  {cpr 4557   “ cima 5621   Fn wfn 6480  ⟶wf 6481  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356  Vtxcvtx 29083  Edgcedg 29134   ClNeighbVtx cclnbgr 48309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-f1o 6492  df-fv 6493

This theorem is referenced by:  grlimgrtri  48494