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Mirrors > Home > MPE Home > Th. List > iunxsn | Structured version Visualization version GIF version |
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.) |
Ref | Expression |
---|---|
iunxsn.1 | ⊢ 𝐴 ∈ V |
iunxsn.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxsn | ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | iunxsn.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | iunxsng 4538 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-sn 4126 df-iun 4457 |
This theorem is referenced by: iunsuc 5724 funopsn 6319 fparlem3 7166 fparlem4 7167 iunfi 8137 kmlem11 8865 ackbij1lem8 8932 dfid6 13616 fsum2dlem 14343 fsumiun 14394 fprod2dlem 14549 prmreclem4 15461 fiuncmp 21017 ovolfiniun 23076 finiunmbl 23119 volfiniun 23122 voliunlem1 23125 iuninc 28761 cvmliftlem10 30530 mrsubvrs 30673 dfrcl4 36987 iunrelexp0 37013 corclrcl 37018 cotrcltrcl 37036 trclfvdecomr 37039 dfrtrcl4 37049 corcltrcl 37050 cotrclrcl 37053 |
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