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Theorem fparlem2 7165
 Description: Lemma for fpar 7168. (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 6117 . . . . . 6 (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd𝑥))
21eqeq1d 2612 . . . . 5 (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd𝑥) = 𝑦))
3 vex 3176 . . . . . . 7 𝑦 ∈ V
43elsn2 4158 . . . . . 6 ((2nd𝑥) ∈ {𝑦} ↔ (2nd𝑥) = 𝑦)
5 fvex 6113 . . . . . . 7 (1st𝑥) ∈ V
65biantrur 526 . . . . . 6 ((2nd𝑥) ∈ {𝑦} ↔ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦}))
74, 6bitr3i 265 . . . . 5 ((2nd𝑥) = 𝑦 ↔ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦}))
82, 7syl6bb 275 . . . 4 (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
98pm5.32i 667 . . 3 ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
10 f2ndres 7082 . . . 4 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 5958 . . . 4 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fniniseg 6246 . . . 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 7092 . . 3 (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ V ∧ (2nd𝑥) ∈ {𝑦})))
159, 13, 143bitr4i 291 . 2 (𝑥 ∈ ((2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦}))
1615eqriv 2607 1 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  fparlem4  7167
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