MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  strfv2 Structured version   Visualization version   GIF version

Theorem strfv2 16533
Description: A variation on strfv 16534 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 16532 1 (𝐶𝑉𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3481  cop 4557  ccnv 5542  Fun wfun 6338  cfv 6344  ndxcnx 16483  Slot cslot 16485
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-res 5555  df-iota 6303  df-fun 6346  df-fv 6352  df-slot 16490
This theorem is referenced by:  strfv  16534
  Copyright terms: Public domain W3C validator