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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 2827 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶)) |
| 3 | 2 | rspcev 3622 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) |
| 4 | eliun 4995 | . 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 1540 ∈ wcel 2108 ∃wrex 3070 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-iun 4993 |
| This theorem is referenced by: oeordi 8625 fseqdom 10066 cfsmolem 10310 axdc3lem2 10491 prmreclem5 16958 efgs1b 19754 lbsextlem2 21161 pmatcoe1fsupp 22707 vitalilem2 25644 weiunse 36469 grpods 42195 oacl2g 43343 omcl2 43346 ofoafg 43367 cnrefiisplem 45844 |
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