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Mirrors > Home > MPE Home > Th. List > eliuni | Structured version Visualization version GIF version |
Description: Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.) |
Ref | Expression |
---|---|
eliuni.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
eliuni | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliuni.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
2 | 1 | eleq2d 2824 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶)) |
3 | 2 | rspcev 3621 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) |
4 | eliun 4999 | . 2 ⊢ (𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) | |
5 | 3, 4 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∪ ciun 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-v 3479 df-iun 4997 |
This theorem is referenced by: oeordi 8623 fseqdom 10063 cfsmolem 10307 axdc3lem2 10488 prmreclem5 16953 efgs1b 19768 lbsextlem2 21178 pmatcoe1fsupp 22722 vitalilem2 25657 weiunse 36450 grpods 42175 oacl2g 43319 omcl2 43322 ofoafg 43343 cnrefiisplem 45784 |
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