MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1stres Structured version   Visualization version   GIF version

Theorem f1stres 8036
Description: Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f1stres (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴

Proof of Theorem f1stres
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3481 . . . . . . . 8 𝑦 ∈ V
2 vex 3481 . . . . . . . 8 𝑧 ∈ V
31, 2op1sta 6246 . . . . . . 7 dom {⟨𝑦, 𝑧⟩} = 𝑦
43eleq1i 2829 . . . . . 6 ( dom {⟨𝑦, 𝑧⟩} ∈ 𝐴𝑦𝐴)
54biimpri 228 . . . . 5 (𝑦𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
65adantr 480 . . . 4 ((𝑦𝐴𝑧𝐵) → dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
76rgen2 3196 . . 3 𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8 sneq 4640 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98dmeqd 5918 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
109unieqd 4924 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
1110eleq1d 2823 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( dom {𝑥} ∈ 𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
1211ralxp 5854 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ ∀𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
137, 12mpbir 231 . 2 𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴
14 df-1st 8012 . . . . 5 1st = (𝑥 ∈ V ↦ dom {𝑥})
1514reseq1i 5995 . . . 4 (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 4019 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 6056 . . . . 5 ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥}))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
1915, 18eqtri 2762 . . 3 (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
2019fmpt 7129 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
2113, 20mpbi 230 1 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  {csn 4630  cop 4636   cuni 4911  cmpt 5230   × cxp 5686  dom cdm 5688  cres 5690  wf 6558  1st c1st 8010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564  df-fn 6565  df-f 6566  df-1st 8012
This theorem is referenced by:  fo1stres  8038  1stcof  8042  fparlem1  8135  domssex2  9175  domssex  9176  unxpwdom2  9625  1stfcl  18252  tx1cn  23632  xpinpreima  33866  xpinpreima2  33867  1stmbfm  34241  hausgraph  43193
  Copyright terms: Public domain W3C validator