Theorem iuneqconst 5008

Description: Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)

Hypothesis

Ref	Expression
iuneqconst.p	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iuneqconst	⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑋

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iuneqconst

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 5001	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		iuneqconst.p	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3	2	eleq2d 2818	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
4	3	rspcev 3612	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	4	adantlr 712	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6	5	ex 412	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
7		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ 𝐴
8		nfra1 3280	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 = 𝐶
9	7, 8	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
10		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
11		rsp 3243	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶))
12		eleq2 2821	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
13	12	biimpd 228	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
14	11, 13	syl6 35	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)))
15	14	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)))
16	9, 10, 15	rexlimd 3262	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
17	6, 16	impbid 211	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
18	1, 17	bitr4id 290	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐶))
19	18	eqrdv 2729	1 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-iun 4999

This theorem is referenced by:  uniimafveqt  46348  imasetpreimafvbijlemfv  46369