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Theorem riinm 3776
Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
riinm ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riinm
StepHypRef Expression
1 incom 3176 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2m 3350 . . . . 5 ((∃𝑥 𝑥𝑋 ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 264 . . . 4 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 3755 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 14 . . 3 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑥𝑋 𝑆𝐴)
6 df-ss 2997 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 120 . 2 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7syl5eq 2127 1 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wral 2353  wrex 2354  cin 2983  wss 2984   ciin 3705
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-ss 2997  df-iin 3707
This theorem is referenced by:  riinerm  6295
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