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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 2875 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ 𝐵 ↔ 𝐸 ∈ 𝐶)) |
3 | 2 | rspcev 3571 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) |
4 | eliun 4885 | . 2 ⊢ (𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝐸 ∈ 𝐵) | |
5 | 3, 4 | sylibr 237 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∪ ciun 4881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-iun 4883 |
This theorem is referenced by: oeordi 8196 fseqdom 9437 cfsmolem 9681 axdc3lem2 9862 prmreclem5 16246 efgs1b 18854 lbsextlem2 19924 pmatcoe1fsupp 21306 vitalilem2 24213 cnrefiisplem 42471 |
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