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Theorem iinss1 3913
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 3234 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝑦𝐶 → ∀𝑥𝐴 𝑦𝐶))
2 vex 2755 . . . 4 𝑦 ∈ V
3 eliin 3906 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
42, 3ax-mp 5 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶)
5 eliin 3906 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
62, 5ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
71, 4, 63imtr4g 205 . 2 (𝐴𝐵 → (𝑦 𝑥𝐵 𝐶𝑦 𝑥𝐴 𝐶))
87ssrdv 3176 1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2160  wral 2468  Vcvv 2752  wss 3144   ciin 3902
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-iin 3904
This theorem is referenced by: (None)
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