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

Theorem strfv2 17080
Description: A variation on strfv 17081 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 (πΈβ€˜ndx)
strfv2.n ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆
Assertion
Ref Expression
strfv2 (𝐢 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘†))

Proof of Theorem strfv2
StepHypRef Expression
1 strfv2.e . 2 𝐸 = Slot (πΈβ€˜ndx)
2 strfv2.s . . 3 𝑆 ∈ V
32a1i 11 . 2 (𝐢 ∈ 𝑉 β†’ 𝑆 ∈ V)
4 strfv2.f . . 3 Fun ◑◑𝑆
54a1i 11 . 2 (𝐢 ∈ 𝑉 β†’ Fun ◑◑𝑆)
6 strfv2.n . . 3 ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆
76a1i 11 . 2 (𝐢 ∈ 𝑉 β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)
8 id 22 . 2 (𝐢 ∈ 𝑉 β†’ 𝐢 ∈ 𝑉)
91, 3, 5, 7, 8strfv2d 17079 1 (𝐢 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444  βŸ¨cop 4593  β—‘ccnv 5633  Fun wfun 6491  β€˜cfv 6497  Slot cslot 17058  ndxcnx 17070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-slot 17059
This theorem is referenced by:  strfv  17081
  Copyright terms: Public domain W3C validator