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Theorem iinss 4759
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iinss (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iinss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3386 . . . 4 𝑦 ∈ V
2 eliin 4713 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 ssel 3790 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
54reximi 3189 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 r19.36v 3264 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
75, 6syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 7syl5bi 234 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
98ssrdv 3802 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  wral 3087  wrex 3088  Vcvv 3383  wss 3767   ciin 4709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-v 3385  df-in 3774  df-ss 3781  df-iin 4711
This theorem is referenced by:  riinn0  4783  reliin  5442  cnviin  5889  iiner  8055  scott0  8997  cfslb  9374  ptbasfi  21710  iscmet3  23416  fnemeet1  32865  pmapglb2N  35784  pmapglb2xN  35785  iinssd  40059  iooiinicc  40501  iooiinioc  40515  meaiininclem  41434  iinhoiicclem  41621  smflim  41719  smflimsuplem7  41766
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