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Mirrors > Home > ILE Home > Th. List > iunxsngf | 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.) |
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 3825 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵) | |
2 | rexsns 3570 | . . . 4 ⊢ (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵) | |
3 | iunxsngf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfcri 2276 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶 |
5 | iunxsngf.2 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
6 | 5 | eleq2d 2210 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
7 | 4, 6 | sbciegf 2944 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
8 | 2, 7 | syl5bb 191 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
9 | 1, 8 | syl5bb 191 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ∪ 𝑥 ∈ {𝐴}𝐵 ↔ 𝑦 ∈ 𝐶)) |
10 | 9 | eqrdv 2138 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Ⅎwnfc 2269 ∃wrex 2418 [wsbc 2913 {csn 3532 ∪ ciun 3821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-sn 3538 df-iun 3823 |
This theorem is referenced by: iunfidisj 6842 iuncld 12323 |
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