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Mirrors > Home > MPE Home > Th. List > iunxsng | Structured version Visualization version GIF version |
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) |
Ref | Expression |
---|---|
iunxsng.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxsng | ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 4999 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵) | |
2 | iunxsng.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | eleq2d 2824 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
4 | 3 | rexsng 4680 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
5 | 1, 4 | bitrid 283 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶)) |
6 | 5 | eqrdv 2732 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {csn 4630 ∪ 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-sn 4631 df-iun 4997 |
This theorem is referenced by: iunxsn 5095 iunxprg 5100 iunxunsn 32586 disjiun2 44997 carageniuncllem1 46476 caratheodorylem1 46481 |
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