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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 6958 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = 〈𝐴, (𝐹‘𝐴)〉)) | |
2 | df-fn 6361 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴})) | |
3 | fnsnb.1 | . . . . . . . . . . 11 ⊢ 𝐴 ∈ V | |
4 | 3 | snid 4563 | . . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴} |
5 | eleq2 2819 | . . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴})) | |
6 | 4, 5 | mpbiri 261 | . . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹) |
7 | 6 | anim2i 620 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) |
8 | 2, 7 | sylbi 220 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) |
9 | funfvop 6848 | . . . . . . 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 2818 | . . . . . 6 ⊢ (𝑥 = 〈𝐴, (𝐹‘𝐴)〉 → (𝑥 ∈ 𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
12 | 10, 11 | syl5ibrcom 250 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 = 〈𝐴, (𝐹‘𝐴)〉 → 𝑥 ∈ 𝐹)) |
13 | 1, 12 | impbid 215 | . . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 = 〈𝐴, (𝐹‘𝐴)〉)) |
14 | velsn 4543 | . . . 4 ⊢ (𝑥 ∈ {〈𝐴, (𝐹‘𝐴)〉} ↔ 𝑥 = 〈𝐴, (𝐹‘𝐴)〉) | |
15 | 13, 14 | bitr4di 292 | . . 3 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {〈𝐴, (𝐹‘𝐴)〉})) |
16 | 15 | eqrdv 2734 | . 2 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
17 | fvex 6708 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
18 | 3, 17 | fnsn 6416 | . . 3 ⊢ {〈𝐴, (𝐹‘𝐴)〉} Fn {𝐴} |
19 | fneq1 6448 | . . 3 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹 Fn {𝐴} ↔ {〈𝐴, (𝐹‘𝐴)〉} Fn {𝐴})) | |
20 | 18, 19 | mpbiri 261 | . 2 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → 𝐹 Fn {𝐴}) |
21 | 16, 20 | impbii 212 | 1 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 {csn 4527 〈cop 4533 dom cdm 5536 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: fnprb 7002 |
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