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Theorem elmpocl 5922
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 5733 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2135 . . . . 5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43dmeqi 4700 . . . 4 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 dmoprabss 5807 . . . 4 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵)
64, 5eqsstri 3095 . . 3 dom 𝐹 ⊆ (𝐴 × 𝐵)
71mpofun 5827 . . . . . 6 Fun 𝐹
8 funrel 5098 . . . . . 6 (Fun 𝐹 → Rel 𝐹)
97, 8ax-mp 7 . . . . 5 Rel 𝐹
10 relelfvdm 5407 . . . . 5 ((Rel 𝐹𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩)) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
119, 10mpan 418 . . . 4 (𝑋 ∈ (𝐹‘⟨𝑆, 𝑇⟩) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
12 df-ov 5731 . . . 4 (𝑆𝐹𝑇) = (𝐹‘⟨𝑆, 𝑇⟩)
1311, 12eleq2s 2209 . . 3 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ dom 𝐹)
146, 13sseldi 3061 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → ⟨𝑆, 𝑇⟩ ∈ (𝐴 × 𝐵))
15 opelxp 4529 . 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 1314  wcel 1463  cop 3496   × cxp 4497  dom cdm 4499  Rel wrel 4504  Fun wfun 5075  cfv 5081  (class class class)co 5728  {coprab 5729  cmpo 5730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733
This theorem is referenced by:  elmpocl1  5923  elmpocl2  5924  elovmpo  5925  elpmi  6515  elmapex  6517  pmsspw  6531  ixxssxr  9576  elixx3g  9577  ixxssixx  9578  eliooxr  9603  elfz2  9690  restsspw  11973  restrcl  12179  ssrest  12194  iscn2  12211  limcrcl  12583
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