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Theorem strfv2 17240
Description: A variation on strfv 17241 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 (𝐶𝑉𝐶 = (𝐸𝑆))

Proof of Theorem strfv2
StepHypRef Expression
1 strfv2.e . 2 𝐸 = Slot (𝐸‘ndx)
2 strfv2.s . . 3 𝑆 ∈ V
32a1i 11 . 2 (𝐶𝑉𝑆 ∈ V)
4 strfv2.f . . 3 Fun 𝑆
54a1i 11 . 2 (𝐶𝑉 → Fun 𝑆)
6 strfv2.n . . 3 ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
76a1i 11 . 2 (𝐶𝑉 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
8 id 22 . 2 (𝐶𝑉𝐶𝑉)
91, 3, 5, 7, 8strfv2d 17239 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  cop 4631  ccnv 5683  Fun wfun 6554  cfv 6560  Slot cslot 17219  ndxcnx 17231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-slot 17220
This theorem is referenced by:  strfv  17241
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