Theorem strslfv 11687

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 11649). By virtue of ndxslid 11668, 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 11655	. . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
4	2, 3	mp1i 10	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
5	2	structfun 11661	. . 3 ⊢ Fun ◡◡𝑆
6	5	a1i 9	. 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆)
7		strfv.n	. . 3 ⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
8	1	simpri 112	. . . . 5 ⊢ (𝐸‘ndx) ∈ ℕ
9		opexg 4079	. . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
10	8, 9	mpan 416	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11		snssg 3595	. . . 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 167	. 2 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
14		id 19	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉)
15	1, 4, 6, 13, 14	strslfv2d 11685	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ⊆ wss 3013  {csn 3466  ⟨cop 3469   class class class wbr 3867  ^◡ccnv 4466  Fun wfun 5043  ‘cfv 5049  ℕcn 8520   Struct cstr 11639  ndxcnx 11640  Slot cslot 11642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fv 5057  df-struct 11645  df-slot 11647

This theorem is referenced by:  strslfv3  11688