MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elmpt2cl Structured version   Visualization version   GIF version

Theorem elmpt2cl 6918
Description: If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
elmpt2cl.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elmpt2cl (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem elmpt2cl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elmpt2cl.f . . . . . 6 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 df-mpt2 6695 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2673 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43dmeqi 5357 . . . 4 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 dmoprabss 6784 . . . 4 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
64, 5eqsstri 3668 . . 3 dom 𝐹 ⊆ (𝐴 × 𝐵)
7 elfvdm 6258 . . . 4 (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
8 df-ov 6693 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
97, 8eleq2s 2748 . . 3 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
106, 9sseldi 3634 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
11 opelxp 5180 . 2 (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆𝐴𝑇𝐵))
1210, 11sylib 208 1 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cop 4216   × cxp 5141  dom cdm 5143  cfv 5926  (class class class)co 6690  {coprab 6691  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  elmpt2cl1  6919  elmpt2cl2  6920  elovmpt2  6921  elovmpt2rab  6922  elovmpt2rab1  6923  el2mpt2csbcl  7295  ixxssixx  12227  funcrcl  16570  natrcl  16657  ismhm  17384  isghm  17707  isga  17770  isslw  18069  isrhm  18769  rimrcl  18772  islmhm  19075  iscn2  21090  elflim2  21815  isfcls  21860  isnmhm  22597  limcrcl  23683  ewlkprop  26555  wwlknbp  26790  wspthnp  26799  iscvm  31367  mclsrcl  31584  mgmhmrcl  42106  intop  42164  rnghmrcl  42214  rngimrcl  42222
  Copyright terms: Public domain W3C validator