Theorem iunxsngf 5057

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.) Avoid ax-13 2402. (Revised by GG, 19-May-2023.)

Hypotheses

Ref	Expression
iunxsngf.1	⊢ Ⅎ𝑥𝐶
iunxsngf.2	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem iunxsngf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4959	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵)
2		iunxsngf.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2	nfcri 2915	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
4		iunxsngf.2	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5	4	eleq2d 2847	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
6	3, 5	rexsngf 4637	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
7	1, 6	bitrid 286	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶))
8	7	eqrdv 2759	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  Ⅎwnfc 2908  ∃wrex 3087  {csn 4588  ∪ ciun 4955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rex 3088  df-v 3455  df-sbc 3744  df-sn 4589  df-iun 4957

This theorem is referenced by:  esum2dlem  34448  fiunelros  34530  iunxsnf  45732