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Mirrors > Home > MPE Home > Th. List > iunxsngf | 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.) (Revised by Thierry Arnoux, 2-May-2020.) Avoid ax-13 2389. (Revised by Gino Giotto, 19-May-2023.) |
Ref | Expression |
---|---|
iunxsngf.1 | ⊢ Ⅎ𝑥𝐶 |
iunxsngf.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxsngf | ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 4926 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵) | |
2 | iunxsngf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
3 | 2 | nfcriv 2970 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶 |
4 | iunxsngf.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 4 | eleq2d 2901 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
6 | 3, 5 | rexsngf 4613 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
7 | 1, 6 | syl5bb 285 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶)) |
8 | 7 | eqrdv 2822 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Ⅎwnfc 2964 ∃wrex 3142 {csn 4570 ∪ ciun 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-sbc 3776 df-sn 4571 df-iun 4924 |
This theorem is referenced by: esum2dlem 31355 fiunelros 31437 iunxsnf 41332 |
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