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Theorem riinm 3824
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 3207 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2m 3388 . . . . 5 ((∃𝑥 𝑥𝑋 ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 265 . . . 4 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 3803 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 14 . . 3 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑥𝑋 𝑆𝐴)
6 df-ss 3026 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 121 . 2 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7syl5eq 2139 1 ((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wex 1433  wcel 1445  wral 2370  wrex 2371  cin 3012  wss 3013   ciin 3753
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-iin 3755
This theorem is referenced by:  riinerm  6405
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