Theorem iunxsng 5046

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Hypothesis

Ref	Expression
iunxsng.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iunxsng	⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsng

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4951	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵)
2		iunxsng.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
3	2	eleq2d 2823	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
4	3	rexsng 4634	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	1, 4	bitrid 283	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶))
6	5	eqrdv 2735	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {csn 4581  ∪ ciun 4947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-sn 4582  df-iun 4949

This theorem is referenced by:  iunxsn  5047  iunxprg  5052  iunxunsn  32645  disjiun2  45381  carageniuncllem1  46842  caratheodorylem1  46847