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Theorem elmpocl 7651
Description: If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
elmpocl.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elmpocl (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 elmpocl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elmpocl.f . . . . . 6 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 df-mpo 7417 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2759 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43dmeqi 5904 . . . 4 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 dmoprabss 7514 . . . 4 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
64, 5eqsstri 4016 . . 3 dom 𝐹 ⊆ (𝐴 × 𝐵)
7 elfvdm 6928 . . . 4 (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
8 df-ov 7415 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
97, 8eleq2s 2850 . . 3 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
106, 9sselid 3980 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
11 opelxp 5712 . 2 (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆𝐴𝑇𝐵))
1210, 11sylib 217 1 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  cop 4634   × cxp 5674  dom cdm 5676  cfv 6543  (class class class)co 7412  {coprab 7413  cmpo 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  elmpocl1  7652  elmpocl2  7653  elovmpo  7654  elovmporab  7655  elovmporab1w  7656  elovmporab1  7657  el2mpocsbcl  8074  ixxssixx  13343  funcrcl  17818  natrcl  17906  mgmhmrcl  18620  ismhm  18708  isghm  19131  isga  19197  isslw  19518  rnghmrcl  20330  rngimrcl  20338  isrhm  20370  rimrcl  20374  islmhm  20783  iscn2  22963  elflim2  23689  isfcls  23734  isnmhm  24484  limcrcl  25624  ewlkprop  29128  wwlknbp  29364  wspthnp  29372  iscvm  34549  mclsrcl  34851  intop  46880  naryrcl  47405
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