Theorem iunxsngf 28758

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.)

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 4460	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵)
2		velsn 4141	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	anbi1i 727	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	exbii 1764	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵))
5		df-rex 2902	. . . . 5 ⊢ (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵))
6		sbc5 3427	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵))
7	4, 5, 6	3bitr4i 291	. . . 4 ⊢ (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
8		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝑦
9		iunxsngf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐶
10	8, 9	nfel 2763	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
11		iunxsngf.2	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	eleq2d 2673	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
13	10, 12	sbciegf 3434	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
14	7, 13	syl5bb 271	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
15	1, 14	syl5bb 271	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶))
16	15	eqrdv 2608	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738  ∃wrex 2897  [wsbc 3402  {csn 4125  ∪ ciun 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-sn 4126  df-iun 4457

This theorem is referenced by:  esum2dlem  29481  fiunelros  29564