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Theorem riinm 3971
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 3339 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2m 3521 . . . . 5 ((∃𝑥 𝑥𝑋 ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 268 . . . 4 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 3950 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 14 . . 3 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑥𝑋 𝑆𝐴)
6 df-ss 3154 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 122 . 2 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7eqtrid 2232 1 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wex 1502  wcel 2158  wral 2465  wrex 2466  cin 3140  wss 3141   ciin 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-in 3147  df-ss 3154  df-iin 3901
This theorem is referenced by:  riinerm  6622
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