Theorem dfiun2gOLD 4961

Description: Obsolete version of dfiun2g 4960 as of 11-Dec-2024. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dfiun2gOLD	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem dfiun2gOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfra1 3144	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
2		rspa 3132	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
3		clel3g 3591	. . . . . . 7 ⊢ (𝐵 ∈ 𝐶 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
4	2, 3	syl 17	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
5	1, 4	rexbida 3251	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
6		rexcom4 3233	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
7	5, 6	bitrdi 287	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
8		r19.41v 3276	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
9	8	exbii 1850	. . . . 5 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
10		exancom 1864	. . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
11	9, 10	bitri 274	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
12	7, 11	bitrdi 287	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)))
13		eliun 4928	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
14		eluniab 4854	. . 3 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
15	12, 13, 14	3bitr4g 314	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
16	15	eqrdv 2736	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  ∪ cuni 4839  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-uni 4840  df-iun 4926

This theorem is referenced by: (None)