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Theorem fparlem2 8096
Description: Lemma for fpar 8099. (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 6903 . . . . . 6 (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd𝑥))
21eqeq1d 2728 . . . . 5 (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd𝑥) = 𝑦))
3 vex 3472 . . . . . . 7 𝑦 ∈ V
43elsn2 4662 . . . . . 6 ((2nd𝑥) ∈ {𝑦} ↔ (2nd𝑥) = 𝑦)
5 fvex 6897 . . . . . . 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 7996 . . . 4 (2nd ↾ (V × V)):(V × V)⟶V
11 ffn 6710 . . . 4 ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12 fniniseg 7054 . . . 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 8006 . . 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 2723 1 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  {csn 4623   × cxp 5667  ccnv 5668  cres 5671  cima 5672   Fn wfn 6531  wf 6532  cfv 6536  1st c1st 7969  2nd c2nd 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-1st 7971  df-2nd 7972
This theorem is referenced by:  fparlem4  8098
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