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Theorem elmpocl1 5969
Description: If a two-parameter class is inhabited, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
elmpocl.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elmpocl1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆𝐴)
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 elmpocl1
StepHypRef Expression
1 elmpocl.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21elmpocl 5968 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
32simpld 111 1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  (class class class)co 5774  cmpo 5776
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 4046  ax-pow 4098  ax-pr 4131
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 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779
This theorem is referenced by:  pmresg  6570  iccssico2  9730  elfzoel1  9922  cncfrss  12731  limccl  12797
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