Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iunssf Structured version   Visualization version   GIF version

Theorem iunssf 38785
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 4494 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21sseq1i 3614 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶)
3 abss 3656 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 dfss2 3577 . . . 4 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
54ralbii 2976 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶))
6 ralcom4 3214 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶))
7 iunssf.1 . . . . . 6 𝑥𝐶
87nfcri 2755 . . . . 5 𝑥 𝑦𝐶
98r19.23 3017 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
109albii 1744 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶))
115, 6, 103bitrri 287 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦𝐶) ↔ ∀𝑥𝐴 𝐵𝐶)
122, 3, 113bitri 286 1 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wcel 1987  {cab 2607  wnfc 2748  wral 2908  wrex 2909  wss 3560   ciun 4492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-in 3567  df-ss 3574  df-iun 4494
This theorem is referenced by:  iunmapss  38916
  Copyright terms: Public domain W3C validator