Theorem relresfin 7268

Description: Restricting to a subset of a finite relation's domain gives a finite set. (Contributed by Jim Kingdon, 6-Oct-2026.)

Hypotheses

Ref	Expression
relresfin.e	⊢ (𝜑 → 𝐸 ∈ Fin)
relresfin.r	⊢ (𝜑 → Rel 𝐸)
relresfin.dm	⊢ (𝜑 → dom 𝐸 ∈ Fin)
relresfin.ss	⊢ (𝜑 → 𝐵 ⊆ dom 𝐸)
relresfin.b	⊢ (𝜑 → 𝐵 ∈ Fin)

Assertion

Ref	Expression
relresfin	⊢ (𝜑 → (𝐸 ↾ 𝐵) ∈ Fin)

Proof of Theorem relresfin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relresfin.e	. 2 ⊢ (𝜑 → 𝐸 ∈ Fin)
2		resss 5089	. . 3 ⊢ (𝐸 ↾ 𝐵) ⊆ 𝐸
3	2	a1i 9	. 2 ⊢ (𝜑 → (𝐸 ↾ 𝐵) ⊆ 𝐸)
4		eleq1 2301	. . . . . 6 ⊢ (𝑦 = (1st ‘𝑥) → (𝑦 ∈ 𝐵 ↔ (1st ‘𝑥) ∈ 𝐵))
5	4	dcbid 850	. . . . 5 ⊢ (𝑦 = (1st ‘𝑥) → (DECID 𝑦 ∈ 𝐵 ↔ DECID (1st ‘𝑥) ∈ 𝐵))
6		relresfin.ss	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ dom 𝐸)
7		relresfin.dm	. . . . . . 7 ⊢ (𝜑 → dom 𝐸 ∈ Fin)
8		relresfin.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
9		fissfi 7267	. . . . . . 7 ⊢ ((𝐵 ⊆ dom 𝐸 ∧ dom 𝐸 ∈ Fin ∧ 𝐵 ∈ Fin) → ∀𝑦 ∈ dom 𝐸DECID 𝑦 ∈ 𝐵)
10	6, 7, 8, 9	syl3anc 1278	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ dom 𝐸DECID 𝑦 ∈ 𝐵)
11	10	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → ∀𝑦 ∈ dom 𝐸DECID 𝑦 ∈ 𝐵)
12		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → 𝑥 ∈ 𝐸)
13		relresfin.r	. . . . . . . . 9 ⊢ (𝜑 → Rel 𝐸)
14		1st2nd 6419	. . . . . . . . 9 ⊢ ((Rel 𝐸 ∧ 𝑥 ∈ 𝐸) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
15	13, 14	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
16	15	eleq1d 2307	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → (𝑥 ∈ 𝐸 ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐸))
17	12, 16	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐸)
18		1stexg 6405	. . . . . . . 8 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V)
19	18	elv 2825	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
20		2ndexg 6406	. . . . . . . 8 ⊢ (𝑥 ∈ V → (2nd ‘𝑥) ∈ V)
21	20	elv 2825	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
22	19, 21	opeldm 4986	. . . . . 6 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐸 → (1st ‘𝑥) ∈ dom 𝐸)
23	17, 22	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → (1st ‘𝑥) ∈ dom 𝐸)
24	5, 11, 23	rspcdva 2934	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → DECID (1st ‘𝑥) ∈ 𝐵)
25	17	biantrurd 305	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → ((1st ‘𝑥) ∈ 𝐵 ↔ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐸 ∧ (1st ‘𝑥) ∈ 𝐵)))
26	21	opelres 5070	. . . . . . 7 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐸 ↾ 𝐵) ↔ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐸 ∧ (1st ‘𝑥) ∈ 𝐵))
27	25, 26	bitr4di 198	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → ((1st ‘𝑥) ∈ 𝐵 ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐸 ↾ 𝐵)))
28	15	eleq1d 2307	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → (𝑥 ∈ (𝐸 ↾ 𝐵) ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐸 ↾ 𝐵)))
29	27, 28	bitr4d 191	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → ((1st ‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (𝐸 ↾ 𝐵)))
30	29	dcbid 850	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → (DECID (1st ‘𝑥) ∈ 𝐵 ↔ DECID 𝑥 ∈ (𝐸 ↾ 𝐵)))
31	24, 30	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → DECID 𝑥 ∈ (𝐸 ↾ 𝐵))
32	31	ralrimiva 2623	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐸 DECID 𝑥 ∈ (𝐸 ↾ 𝐵))
33		ssfidc 7249	. 2 ⊢ ((𝐸 ∈ Fin ∧ (𝐸 ↾ 𝐵) ⊆ 𝐸 ∧ ∀𝑥 ∈ 𝐸 DECID 𝑥 ∈ (𝐸 ↾ 𝐵)) → (𝐸 ↾ 𝐵) ∈ Fin)
34	1, 3, 32, 33	syl3anc 1278	1 ⊢ (𝜑 → (𝐸 ↾ 𝐵) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  ⟨cop 3712  dom cdm 4776   ↾ cres 4778  Rel wrel 4781  ‘cfv 5380  1^st c1st 6376  2^nd c2nd 6377  Fincfn 7026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029

This theorem is used by:  gsum2dfilem2  14272