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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvsingle | Structured version Visualization version GIF version |
Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.) |
Ref | Expression |
---|---|
fvsingle | ⊢ (Singleton‘𝐴) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴)) | |
2 | sneq 4187 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 1, 2 | eqeq12d 2637 | . . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴})) |
4 | eqid 2622 | . . . . 5 ⊢ {𝑥} = {𝑥} | |
5 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | snex 4908 | . . . . . 6 ⊢ {𝑥} ∈ V | |
7 | 5, 6 | brsingle 32024 | . . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥}) |
8 | 4, 7 | mpbir 221 | . . . 4 ⊢ 𝑥Singleton{𝑥} |
9 | fnsingle 32026 | . . . . 5 ⊢ Singleton Fn V | |
10 | fnbrfvb 6236 | . . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})) | |
11 | 9, 5, 10 | mp2an 708 | . . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}) |
12 | 8, 11 | mpbir 221 | . . 3 ⊢ (Singleton‘𝑥) = {𝑥} |
13 | 3, 12 | vtoclg 3266 | . 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
14 | fvprc 6185 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅) | |
15 | snprc 4253 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
16 | 15 | biimpi 206 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
17 | 14, 16 | eqtr4d 2659 | . 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
18 | 13, 17 | pm2.61i 176 | 1 ⊢ (Singleton‘𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 {csn 4177 class class class wbr 4653 Fn wfn 5883 ‘cfv 5888 Singletoncsingle 31945 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-symdif 3844 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-eprel 5029 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-2nd 7169 df-txp 31961 df-singleton 31969 |
This theorem is referenced by: (None) |
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