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| Mirrors > Home > MPE Home > Th. List > fnsnb | Structured version Visualization version GIF version | ||
| Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.) | 
| Ref | Expression | 
|---|---|
| fnsnb.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| fnsnb | ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnsnr 7185 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = 〈𝐴, (𝐹‘𝐴)〉)) | |
| 2 | df-fn 6564 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴})) | |
| 3 | fnsnb.1 | . . . . . . . . . . 11 ⊢ 𝐴 ∈ V | |
| 4 | 3 | snid 4662 | . . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴} | 
| 5 | eleq2 2830 | . . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴})) | |
| 6 | 4, 5 | mpbiri 258 | . . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹) | 
| 7 | 6 | anim2i 617 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) | 
| 8 | 2, 7 | sylbi 217 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) | 
| 9 | funfvop 7070 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
| 10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | 
| 11 | eleq1 2829 | . . . . . 6 ⊢ (𝑥 = 〈𝐴, (𝐹‘𝐴)〉 → (𝑥 ∈ 𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
| 12 | 10, 11 | syl5ibrcom 247 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 = 〈𝐴, (𝐹‘𝐴)〉 → 𝑥 ∈ 𝐹)) | 
| 13 | 1, 12 | impbid 212 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 = 〈𝐴, (𝐹‘𝐴)〉)) | 
| 14 | velsn 4642 | . . . 4 ⊢ (𝑥 ∈ {〈𝐴, (𝐹‘𝐴)〉} ↔ 𝑥 = 〈𝐴, (𝐹‘𝐴)〉) | |
| 15 | 13, 14 | bitr4di 289 | . . 3 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {〈𝐴, (𝐹‘𝐴)〉})) | 
| 16 | 15 | eqrdv 2735 | . 2 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) | 
| 17 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
| 18 | 3, 17 | fnsn 6624 | . . 3 ⊢ {〈𝐴, (𝐹‘𝐴)〉} Fn {𝐴} | 
| 19 | fneq1 6659 | . . 3 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹 Fn {𝐴} ↔ {〈𝐴, (𝐹‘𝐴)〉} Fn {𝐴})) | |
| 20 | 18, 19 | mpbiri 258 | . 2 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → 𝐹 Fn {𝐴}) | 
| 21 | 16, 20 | impbii 209 | 1 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 | 
| 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-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 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-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 | 
| This theorem is referenced by: fnprb 7228 | 
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