![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4975 | . 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 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ∪ ciun 4881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-sn 4526 df-iun 4883 |
This theorem is referenced by: iunsuc 6241 funopsn 6887 fparlem3 7792 fparlem4 7793 iunfi 8796 kmlem11 9571 ackbij1lem8 9638 dfid6 14379 fsum2dlem 15117 fsumiun 15168 fprod2dlem 15326 prmreclem4 16245 fiuncmp 22009 ovolfiniun 24105 finiunmbl 24148 volfiniun 24151 voliunlem1 24154 iuninc 30324 cvmliftlem10 32654 mrsubvrs 32882 dfrcl4 40377 iunrelexp0 40403 corclrcl 40408 cotrcltrcl 40426 trclfvdecomr 40429 dfrtrcl4 40439 corcltrcl 40440 cotrclrcl 40443 |
Copyright terms: Public domain | W3C validator |