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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 6921 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = 〈𝐴, (𝐹‘𝐴)〉)) | |
2 | df-fn 6352 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴})) | |
3 | fnsnb.1 | . . . . . . . . . . 11 ⊢ 𝐴 ∈ V | |
4 | 3 | snid 4594 | . . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴} |
5 | eleq2 2901 | . . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴})) | |
6 | 4, 5 | mpbiri 260 | . . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹) |
7 | 6 | anim2i 618 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) |
8 | 2, 7 | sylbi 219 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) |
9 | funfvop 6814 | . . . . . . 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 2900 | . . . . . 6 ⊢ (𝑥 = 〈𝐴, (𝐹‘𝐴)〉 → (𝑥 ∈ 𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
12 | 10, 11 | syl5ibrcom 249 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 = 〈𝐴, (𝐹‘𝐴)〉 → 𝑥 ∈ 𝐹)) |
13 | 1, 12 | impbid 214 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 = 〈𝐴, (𝐹‘𝐴)〉)) |
14 | velsn 4576 | . . . 4 ⊢ (𝑥 ∈ {〈𝐴, (𝐹‘𝐴)〉} ↔ 𝑥 = 〈𝐴, (𝐹‘𝐴)〉) | |
15 | 13, 14 | syl6bbr 291 | . . 3 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {〈𝐴, (𝐹‘𝐴)〉})) |
16 | 15 | eqrdv 2819 | . 2 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
17 | fvex 6677 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
18 | 3, 17 | fnsn 6406 | . . 3 ⊢ {〈𝐴, (𝐹‘𝐴)〉} Fn {𝐴} |
19 | fneq1 6438 | . . 3 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹 Fn {𝐴} ↔ {〈𝐴, (𝐹‘𝐴)〉} Fn {𝐴})) | |
20 | 18, 19 | mpbiri 260 | . 2 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → 𝐹 Fn {𝐴}) |
21 | 16, 20 | impbii 211 | 1 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 〈cop 4566 dom cdm 5549 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: fnprb 6965 |
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