Theorem strfv2 16522

Description: A variation on strfv 16523 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.)

Hypotheses

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

Assertion

Ref	Expression
strfv2	⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Proof of Theorem strfv2

Step	Hyp	Ref	Expression
1		strfv2.e	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
2		strfv2.s	. . 3 ⊢ 𝑆 ∈ V
3	2	a1i 11	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V)
4		strfv2.f	. . 3 ⊢ Fun ◡◡𝑆
5	4	a1i 11	. 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆)
6		strfv2.n	. . 3 ⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
7	6	a1i 11	. 2 ⊢ (𝐶 ∈ 𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
8		id 22	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉)
9	1, 3, 5, 7, 8	strfv2d 16521	1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531  ^◡ccnv 5518  Fun wfun 6318  ‘cfv 6324  ndxcnx 16472  Slot cslot 16474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-slot 16479

This theorem is referenced by:  strfv  16523