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Theorem riinn0 4525
Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinn0 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riinn0
StepHypRef Expression
1 incom 3766 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2z 4011 . . . . 5 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 467 . . . 4 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 4501 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 17 . . 3 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → 𝑥𝑋 𝑆𝐴)
6 df-ss 3553 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 206 . 2 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7syl5eq 2655 1 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wne 2779  wral 2895  wrex 2896  cin 3538  wss 3539  c0 3873   ciin 4450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-iin 4452
This theorem is referenced by:  riinrab  4526  riiner  7684  mreriincl  16027  riinopn  20480  alexsublem  21600  fnemeet1  31337
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