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Theorem elmpt2cl 5776
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 5595 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2103 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43dmeqi 4594 . . . 4 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 dmoprabss 5664 . . . 4 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
64, 5eqsstri 3040 . . 3 dom 𝐹 ⊆ (𝐴 × 𝐵)
71mpt2fun 5681 . . . . . 6 Fun 𝐹
8 funrel 4985 . . . . . 6 (Fun 𝐹 → Rel 𝐹)
97, 8ax-mp 7 . . . . 5 Rel 𝐹
10 relelfvdm 5280 . . . . 5 ((Rel 𝐹𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
119, 10mpan 415 . . . 4 (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
12 df-ov 5593 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
1311, 12eleq2s 2177 . . 3 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
146, 13sseldi 3008 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
15 opelxp 4429 . 2 (⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵) ↔ (𝑆𝐴𝑇𝐵))
1614, 15sylib 120 1 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  cop 3425   × cxp 4398  dom cdm 4400  Rel wrel 4405  Fun wfun 4962  cfv 4968  (class class class)co 5590  {coprab 5591  cmpt2 5592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4083  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-iota 4933  df-fun 4970  df-fv 4976  df-ov 5593  df-oprab 5594  df-mpt2 5595
This theorem is referenced by:  elmpt2cl1  5777  elmpt2cl2  5778  elovmpt2  5779  elpmi  6353  elmapex  6355  pmsspw  6369  ixxssxr  9212  elixx3g  9213  ixxssixx  9214  eliooxr  9239  elfz2  9325
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