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Theorem fparlem1 7796
Description: Lemma for fpar 7800. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem1 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Proof of Theorem fparlem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvres 6682 . . . . . 6 (𝑦 ∈ (V × V) → ((1st ↾ (V × V))‘𝑦) = (1st𝑦))
21eqeq1d 2820 . . . . 5 (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1st𝑦) = 𝑥))
3 vex 3495 . . . . . . 7 𝑥 ∈ V
43elsn2 4594 . . . . . 6 ((1st𝑦) ∈ {𝑥} ↔ (1st𝑦) = 𝑥)
5 fvex 6676 . . . . . . 7 (2nd𝑦) ∈ V
65biantru 530 . . . . . 6 ((1st𝑦) ∈ {𝑥} ↔ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V))
74, 6bitr3i 278 . . . . 5 ((1st𝑦) = 𝑥 ↔ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V))
82, 7syl6bb 288 . . . 4 (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V)))
98pm5.32i 575 . . 3 ((𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V)))
10 f1stres 7702 . . . 4 (1st ↾ (V × V)):(V × V)⟶V
11 ffn 6507 . . . 4 ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
12 fniniseg 6822 . . . 4 ((1st ↾ (V × V)) Fn (V × V) → (𝑦 ∈ ((1st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥)))
1310, 11, 12mp2b 10 . . 3 (𝑦 ∈ ((1st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥))
14 elxp7 7713 . . 3 (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V)))
159, 13, 143bitr4i 304 . 2 (𝑦 ∈ ((1st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V))
1615eqriv 2815 1 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557   × cxp 5546  ccnv 5547  cres 5550  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-1st 7678  df-2nd 7679
This theorem is referenced by:  fparlem3  7798
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