MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fparlem1 Structured version   Visualization version   GIF version

Theorem fparlem1 8123
Description: Lemma for fpar 8127. (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 6921 . . . . . 6 (𝑦 ∈ (V × V) → ((1st ↾ (V × V))‘𝑦) = (1st𝑦))
21eqeq1d 2730 . . . . 5 (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1st𝑦) = 𝑥))
3 vex 3477 . . . . . . 7 𝑥 ∈ V
43elsn2 4672 . . . . . 6 ((1st𝑦) ∈ {𝑥} ↔ (1st𝑦) = 𝑥)
5 fvex 6915 . . . . . . 7 (2nd𝑦) ∈ V
65biantru 528 . . . . . 6 ((1st𝑦) ∈ {𝑥} ↔ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V))
74, 6bitr3i 276 . . . . 5 ((1st𝑦) = 𝑥 ↔ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V))
82, 7bitrdi 286 . . . 4 (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V)))
98pm5.32i 573 . . 3 ((𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V)))
10 f1stres 8023 . . . 4 (1st ↾ (V × V)):(V × V)⟶V
11 ffn 6727 . . . 4 ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V))
12 fniniseg 7074 . . . 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 8034 . . 3 (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1st𝑦) ∈ {𝑥} ∧ (2nd𝑦) ∈ V)))
159, 13, 143bitr4i 302 . 2 (𝑦 ∈ ((1st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V))
1615eqriv 2725 1 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  {csn 4632   × cxp 5680  ccnv 5681  cres 5684  cima 5685   Fn wfn 6548  wf 6549  cfv 6553  1st c1st 7997  2nd c2nd 7998
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-1st 7999  df-2nd 8000
This theorem is referenced by:  fparlem3  8125
  Copyright terms: Public domain W3C validator