Theorem strslfv2 12665

Description: A variation on strslfv 12666 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv2.s	⊢ 𝑆 ∈ V
strfv2.f	⊢ Fun ◡◡𝑆
strslfv2.e	⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
strfv2.n	⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆

Assertion

Ref	Expression
strslfv2	⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Proof of Theorem strslfv2

Step	Hyp	Ref	Expression
1		strslfv2.e	. 2 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
2		strfv2.s	. . 3 ⊢ 𝑆 ∈ V
3	2	a1i 9	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
4		strfv2.f	. . 3 ⊢ Fun ◡◡𝑆
5	4	a1i 9	. 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆)
6		strfv2.n	. . 3 ⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
7	6	a1i 9	. 2 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
8		id 19	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉)
9	1, 3, 5, 7, 8	strslfv2d 12664	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ⟨cop 3622  ^◡ccnv 4659  Fun wfun 5249  ‘cfv 5255  ℕcn 8984  ndxcnx 12618  Slot cslot 12620

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-slot 12625

This theorem is referenced by: (None)