Theorem strslfv2 12722

Description: A variation on strslfv 12723 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 12721	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763  ⟨cop 3625  ^◡ccnv 4662  Fun wfun 5252  ‘cfv 5258  ℕcn 8990  ndxcnx 12675  Slot cslot 12677

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 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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  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-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-slot 12682

This theorem is referenced by: (None)