MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunssf Structured version   Visualization version   GIF version

Theorem iunssf 4953
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 4906 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21sseq1i 3929 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶)
3 abss 3974 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 dfss2 3886 . . . 4 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
54ralbii 3088 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶))
6 ralcom4 3157 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶))
7 iunssf.1 . . . . . 6 𝑥𝐶
87nfcri 2891 . . . . 5 𝑥 𝑦𝐶
98r19.23 3233 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
109albii 1827 . . 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 1541  wcel 2110  {cab 2714  wnfc 2884  wral 3061  wrex 3062  wss 3866   ciun 4904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-v 3410  df-in 3873  df-ss 3883  df-iun 4906
This theorem is referenced by:  djussxp2  30704  iunmapss  42428
  Copyright terms: Public domain W3C validator