Theorem iunconstm 3906

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 3902	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.9rmv 3526	. . 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 2185	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1363  ∃wex 1502   ∈ wcel 2158  ∃wrex 2466  ∪ ciun 3898

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-iun 3900

This theorem is referenced by: (None)