Theorem resseqnbasd 13307

Description: The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)

Hypotheses

Ref	Expression
resseqnbas.r	⊢ 𝑅 = (𝑊 ↾s 𝐴)
resseqnbas.e	⊢ 𝐶 = (𝐸‘𝑊)
resseqnbasd.f	⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
resseqnbas.n	⊢ (𝐸‘ndx) ≠ (Base‘ndx)
resseqnbasd.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)
resseqnbasd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
resseqnbasd	⊢ (𝜑 → 𝐶 = (𝐸‘𝑅))

Proof of Theorem resseqnbasd

Step	Hyp	Ref	Expression
1		resseqnbas.e	. 2 ⊢ 𝐶 = (𝐸‘𝑊)
2		resseqnbas.r	. . . . 5 ⊢ 𝑅 = (𝑊 ↾s 𝐴)
3		resseqnbasd.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
4		resseqnbasd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		ressvalsets 13298	. . . . . 6 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
6	3, 4, 5	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
7	2, 6	eqtrid 2279	. . . 4 ⊢ (𝜑 → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8	7	fveq2d 5676	. . 3 ⊢ (𝜑 → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)))
9		inex1g 4248	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
10	4, 9	syl 14	. . . 4 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
11		resseqnbasd.f	. . . . 5 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
12		resseqnbas.n	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Base‘ndx)
13		basendxnn 13289	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
14	11, 12, 13	setsslnid 13285	. . . 4 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝐴 ∩ (Base‘𝑊)) ∈ V) → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)))
15	3, 10, 14	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)))
16	8, 15	eqtr4d 2270	. 2 ⊢ (𝜑 → (𝐸‘𝑅) = (𝐸‘𝑊))
17	1, 16	eqtr4id 2286	1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  Vcvv 2815   ∩ cin 3212  ⟨cop 3694  ‘cfv 5354  (class class class)co 6052  ℕcn 9242  ndxcnx 13230   sSet csts 13231  Slot cslot 13232  Basecbs 13233   ↾_s cress 13234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1re 8226  ax-addrcl 8229

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-inn 9243  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241

This theorem is referenced by:  ressplusgd  13363  ressmulrg  13379  ressscag  13417  ressvscag  13418  ressipg  13419