Theorem strslfv 13346

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 13307). By virtue of ndxslid 13326, 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 13313	. . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
4	2, 3	mp1i 10	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
5	2	structfun 13319	. . 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 4350	. . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
10	8, 9	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11		snssg 3834	. . . 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 13344	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  {csn 3695  ⟨cop 3698   class class class wbr 4115  ^◡ccnv 4754  Fun wfun 5352  ‘cfv 5358  ℕcn 9258   Struct cstr 13297  ndxcnx 13298  Slot cslot 13300

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 4234  ax-pow 4293  ax-pr 4328  ax-un 4560

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 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-iota 5318  df-fun 5360  df-fv 5366  df-struct 13303  df-slot 13305

This theorem is referenced by:  cnfldbas  14839  mpocnfldadd  14840  mpocnfldmul  14842  cnfldcj  14844  cnfldtset  14845  cnfldle  14846  cnfldds  14847