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Theorem iunss 3726
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 3687 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21sseq1i 2997 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶)
3 abss 3037 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 dfss2 2962 . . . 4 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
54ralbii 2347 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶))
6 ralcom4 2593 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶))
7 r19.23v 2442 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
87albii 1375 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
95, 6, 83bitrri 200 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶) ↔ ∀𝑥𝐴 𝐵𝐶)
102, 3, 93bitri 199 1 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wcel 1409  {cab 2042  wral 2323  wrex 2324  wss 2945   ciun 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-iun 3687
This theorem is referenced by:  iunss2  3730  djussxp  4509  fun11iun  5175
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