Theorem eliuni 4963

Description: Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
eliuni	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eliuni

Step	Hyp	Ref	Expression
1		eliuni.1	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
2	1	eleq2d 2849	. . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶))
3	2	rspcev 3582	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵)
4		eliun 4961	. 2 ⊢ (𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵)
5	3, 4	sylibr 237	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ∪ ciun 4957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3457  df-iun 4959

This theorem is referenced by:  oeordi  8574  fseqdom  10011  cfsmolem  10255  axdc3lem2  10436  prmreclem5  16981  efgs1b  19807  lbsextlem2  21264  pmatcoe1fsupp  22839  vitalilem2  25749  weiunse  36957  ttcid  36981  grpods  42939  oacl2g  44037  omcl2  44040  ofoafg  44061  cnrefiisplem  46523