MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fparlem2 Structured version   Visualization version   GIF version

Theorem fparlem2 8094
Description: Lemma for fpar 8097. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem2 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Proof of Theorem fparlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvres 6907 . . . . . 6 (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd𝑥))
21eqeq1d 2735 . . . . 5 (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd𝑥) = 𝑦))
3 vex 3479 . . . . . . 7 𝑦 ∈ V
43elsn2 4666 . . . . . 6 ((2nd𝑥) ∈ {𝑦} ↔ (2nd𝑥) = 𝑦)
5 fvex 6901 . . . . . . 7 (1st𝑥) ∈ V
65biantrur 532 . . . . . 6 ((2nd𝑥) ∈ {𝑦} ↔ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦}))
74, 6bitr3i 277 . . . . 5 ((2nd𝑥) = 𝑦 ↔ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦}))
82, 7bitrdi 287 . . . 4 (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
98pm5.32i 576 . . 3 ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
10 f2ndres 7995 . . . 4 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6714 . . . 4 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fniniseg 7057 . . . 4 ((2nd ↾ (V × V)) Fn (V × V) → (𝑥 ∈ ((2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦)))
1310, 11, 12mp2b 10 . . 3 (𝑥 ∈ ((2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦))
14 elxp7 8005 . . 3 (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
159, 13, 143bitr4i 303 . 2 (𝑥 ∈ ((2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦}))
1615eqriv 2730 1 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  {csn 4627   × cxp 5673  ccnv 5674  cres 5677  cima 5678   Fn wfn 6535  wf 6536  cfv 6540  1st c1st 7968  2nd c2nd 7969
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 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-1st 7970  df-2nd 7971
This theorem is referenced by:  fparlem4  8096
  Copyright terms: Public domain W3C validator