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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 6906 | . . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴)) | |
| 2 | sneq 4636 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
| 3 | 1, 2 | eqeq12d 2753 | . . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴})) |
| 4 | eqid 2737 | . . . . 5 ⊢ {𝑥} = {𝑥} | |
| 5 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 6 | vsnex 5434 | . . . . . 6 ⊢ {𝑥} ∈ V | |
| 7 | 5, 6 | brsingle 35918 | . . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥}) |
| 8 | 4, 7 | mpbir 231 | . . . 4 ⊢ 𝑥Singleton{𝑥} |
| 9 | fnsingle 35920 | . . . . 5 ⊢ Singleton Fn V | |
| 10 | fnbrfvb 6959 | . . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})) | |
| 11 | 9, 5, 10 | mp2an 692 | . . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}) |
| 12 | 8, 11 | mpbir 231 | . . 3 ⊢ (Singleton‘𝑥) = {𝑥} |
| 13 | 3, 12 | vtoclg 3554 | . 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
| 14 | fvprc 6898 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅) | |
| 15 | snprc 4717 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 16 | 15 | biimpi 216 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 17 | 14, 16 | eqtr4d 2780 | . 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
| 18 | 13, 17 | pm2.61i 182 | 1 ⊢ (Singleton‘𝐴) = {𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 Fn wfn 6556 ‘cfv 6561 Singletoncsingle 35839 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-singleton 35863 |
| This theorem is referenced by: (None) |
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