Theorem iunconstm 3935

Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Jim Kingdon, 15-Aug-2018.)

Assertion

Ref	Expression
iunconstm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunconstm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3931	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.9rmv 3552	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
3	1, 2	bitr4id 199	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵))
4	3	eqrdv 2203	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∃wrex 2485  ∪ ciun 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-iun 3929

This theorem is referenced by: (None)