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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 2892 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶)) |
3 | 2 | rspcev 3526 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) |
4 | eliun 4746 | . 2 ⊢ (𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) | |
5 | 3, 4 | sylibr 226 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ∪ ciun 4742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-v 3416 df-iun 4744 |
This theorem is referenced by: oeordi 7939 fseqdom 9169 cfsmolem 9414 axdc3lem2 9595 prmreclem5 16002 efgs1b 18507 lbsextlem2 19527 pmatcoe1fsupp 20883 vitalilem2 23782 cnrefiisplem 40844 |
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