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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 2673 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶)) |
3 | 2 | rspcev 3282 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) |
4 | eliun 4460 | . 2 ⊢ (𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) | |
5 | 3, 4 | sylibr 223 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-iun 4457 |
This theorem is referenced by: oeordi 7554 fseqdom 8732 cfsmolem 8975 axdc3lem2 9156 prmreclem5 15462 efgs1b 17972 lbsextlem2 18980 pmatcoe1fsupp 20325 vitalilem2 23184 numclwwlkun 26606 |
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