Theorem eliuni 4947

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 2819	. . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶))
3	2	rspcev 3573	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵)
4		eliun 4945	. 2 ⊢ (𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵)
5	3, 4	sylibr 234	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  ∪ ciun 4941

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-v 3439  df-iun 4943

This theorem is referenced by:  oeordi  8508  fseqdom  9924  cfsmolem  10168  axdc3lem2  10349  prmreclem5  16834  efgs1b  19650  lbsextlem2  21098  pmatcoe1fsupp  22617  vitalilem2  25538  weiunse  36533  grpods  42307  oacl2g  43447  omcl2  43450  ofoafg  43471  cnrefiisplem  45951