Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iunxunpr Structured version   Visualization version   GIF version

Theorem iunxunpr 32580
Description: Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Hypotheses
Ref Expression
iunxunsn.1 (𝑥 = 𝑋𝐵 = 𝐶)
iunxunpr.2 (𝑥 = 𝑌𝐵 = 𝐷)
Assertion
Ref Expression
iunxunpr ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝑥,𝐷   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunxunpr
StepHypRef Expression
1 iunxun 5094 . 2 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋, 𝑌}𝐵)
2 iunxunsn.1 . . . 4 (𝑥 = 𝑋𝐵 = 𝐶)
3 iunxunpr.2 . . . 4 (𝑥 = 𝑌𝐵 = 𝐷)
42, 3iunxprg 5096 . . 3 ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶𝐷))
54uneq2d 4168 . 2 ((𝑋𝑉𝑌𝑊) → ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋, 𝑌}𝐵) = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
61, 5eqtrid 2789 1 ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  {cpr 4628   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-iun 4993
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator