Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunxsngf2 | Structured version Visualization version GIF version |
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
iunxsngf2.1 | ⊢ Ⅎ𝑥𝐶 |
iunxsngf2.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxsngf2 | ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 4524 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵) | |
2 | iunxsngf2.1 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
3 | 2 | nfcri 2758 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶 |
4 | iunxsngf2.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 4 | eleq2d 2687 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
6 | 3, 5 | rexsngf 39220 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
7 | 1, 6 | syl5bb 272 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶)) |
8 | 7 | eqrdv 2620 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∃wrex 2913 {csn 4177 ∪ ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-sn 4178 df-iun 4522 |
This theorem is referenced by: iunxsnf 39233 |
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