Theorem strslfv 12510

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 12471). By virtue of ndxslid 12490, 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 12477	. . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
4	2, 3	mp1i 10	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
5	2	structfun 12483	. . 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 4230	. . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
10	8, 9	mpan 424	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ V)
11		snssg 3728	. . . 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 12508	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ⊆ wss 3131  {csn 3594  ⟨cop 3597   class class class wbr 4005  ^◡ccnv 4627  Fun wfun 5212  ‘cfv 5218  ℕcn 8922   Struct cstr 12461  ndxcnx 12462  Slot cslot 12464

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-struct 12467  df-slot 12469

This theorem is referenced by:  strslfv3  12511  cnfldbas  13599  cnfldadd  13600  cnfldmul  13601  cnfldcj  13602