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Theorem iunss 3914
Description: Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iunss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-iun 3875 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21sseq1i 3173 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶)
3 abss 3216 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 dfss2 3136 . . . 4 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
54ralbii 2476 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶))
6 ralcom4 2752 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶))
7 r19.23v 2579 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
87albii 1463 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
95, 6, 83bitrri 206 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶) ↔ ∀𝑥𝐴 𝐵𝐶)
102, 3, 93bitri 205 1 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346  wcel 2141  {cab 2156  wral 2448  wrex 2449  wss 3121   ciun 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875
This theorem is referenced by:  iunss2  3918  djussxp  4756  fun11iun  5463  ennnfonelemf1  12373  tgidm  12868
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