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Theorem elmpocl 5971
Description: If a two-parameter class is inhabited, 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 5782 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2160 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43dmeqi 4743 . . . 4 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 dmoprabss 5856 . . . 4 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
64, 5eqsstri 3129 . . 3 dom 𝐹 ⊆ (𝐴 × 𝐵)
71mpofun 5876 . . . . . 6 Fun 𝐹
8 funrel 5143 . . . . . 6 (Fun 𝐹 → Rel 𝐹)
97, 8ax-mp 5 . . . . 5 Rel 𝐹
10 relelfvdm 5456 . . . . 5 ((Rel 𝐹𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
119, 10mpan 420 . . . 4 (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
12 df-ov 5780 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
1311, 12eleq2s 2234 . . 3 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
146, 13sseldi 3095 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
15 opelxp 4572 . 2 (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆𝐴𝑇𝐵))
1614, 15sylib 121 1 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  cop 3530   × cxp 4540  dom cdm 4542  Rel wrel 4547  Fun wfun 5120  cfv 5126  (class class class)co 5777  {coprab 5778  cmpo 5779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-iota 5091  df-fun 5128  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782
This theorem is referenced by:  elmpocl1  5972  elmpocl2  5973  elovmpo  5974  elpmi  6564  elmapex  6566  pmsspw  6580  ixxssxr  9706  elixx3g  9707  ixxssixx  9708  eliooxr  9733  elfz2  9821  restsspw  12156  restrcl  12362  ssrest  12377  iscn2  12395  ishmeo  12499  limcrcl  12822
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