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Theorem iinss1 3764
Description: Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssralv 3100 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝑦𝐶 → ∀𝑥𝐴 𝑦𝐶))
2 vex 2636 . . . 4 𝑦 ∈ V
3 eliin 3757 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
42, 3ax-mp 7 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶)
5 eliin 3757 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
62, 5ax-mp 7 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
71, 4, 63imtr4g 204 . 2 (𝐴𝐵 → (𝑦 𝑥𝐵 𝐶𝑦 𝑥𝐴 𝐶))
87ssrdv 3045 1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1445  wral 2370  Vcvv 2633  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-v 2635  df-in 3019  df-ss 3026  df-iin 3755
This theorem is referenced by: (None)
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