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Theorem fparlem2 8137
Description: Lemma for fpar 8140. (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 6926 . . . . . 6 (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd𝑥))
21eqeq1d 2737 . . . . 5 (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd𝑥) = 𝑦))
3 vex 3482 . . . . . . 7 𝑦 ∈ V
43elsn2 4670 . . . . . 6 ((2nd𝑥) ∈ {𝑦} ↔ (2nd𝑥) = 𝑦)
5 fvex 6920 . . . . . . 7 (1st𝑥) ∈ V
65biantrur 530 . . . . . 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 574 . . 3 ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
10 f2ndres 8038 . . . 4 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6737 . . . 4 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fniniseg 7080 . . . 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 8048 . . 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 2732 1 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   × cxp 5687  ccnv 5688  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  fparlem4  8139
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