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Theorem iunssf 4935
 Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
iunssf.1 𝑥𝐶
Assertion
Ref Expression
iunssf ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)

Proof of Theorem iunssf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-iun 4887 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21sseq1i 3945 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶)
3 abss 3990 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 dfss2 3903 . . . 4 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
54ralbii 3133 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶))
6 ralcom4 3198 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶))
7 iunssf.1 . . . . . 6 𝑥𝐶
87nfcri 2943 . . . . 5 𝑥 𝑦𝐶
98r19.23 3274 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
109albii 1821 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
115, 6, 103bitrri 301 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶) ↔ ∀𝑥𝐴 𝐵𝐶)
122, 3, 113bitri 300 1 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   ⊆ wss 3883  ∪ ciun 4885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-in 3890  df-ss 3900  df-iun 4887 This theorem is referenced by:  djussxp2  30454  iunmapss  42014
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