Theorem strslfv 12748

Description: Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12709). By virtue of ndxslid 12728, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv.s	⊢ 𝑆 Struct 𝑋
strslfv.e	⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
strfv.n	⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆

Assertion

Ref	Expression
strslfv	⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Proof of Theorem strslfv

Step	Hyp	Ref	Expression
1		strslfv.e	. 2 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
2		strfv.s	. . 3 ⊢ 𝑆 Struct 𝑋
3		structex 12715	. . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
4	2, 3	mp1i 10	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
5	2	structfun 12721	. . 3 ⊢ Fun ◡◡𝑆
6	5	a1i 9	. 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆)
7		strfv.n	. . 3 ⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
8	1	simpri 113	. . . . 5 ⊢ (𝐸‘ndx) ∈ ℕ
9		opexg 4262	. . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
10	8, 9	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11		snssg 3757	. . . 4 ⊢ (⟨(𝐸‘ndx), 𝐶⟩ ∈ V → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 ↔ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆))
12	10, 11	syl 14	. . 3 ⊢ (𝐶 ∈ 𝑉 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 ↔ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆))
13	7, 12	mpbiri 168	. 2 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
14		id 19	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉)
15	1, 4, 6, 13, 14	strslfv2d 12746	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  {csn 3623  ⟨cop 3626   class class class wbr 4034  ^◡ccnv 4663  Fun wfun 5253  ‘cfv 5259  ℕcn 9007   Struct cstr 12699  ndxcnx 12700  Slot cslot 12702

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-struct 12705  df-slot 12707

This theorem is referenced by:  cnfldbas  14192  mpocnfldadd  14193  mpocnfldmul  14195  cnfldcj  14197  cnfldtset  14198  cnfldle  14199  cnfldds  14200