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Mirrors > Home > MPE Home > Th. List > fsn2g | Structured version Visualization version GIF version |
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
Ref | Expression |
---|---|
fsn2g | ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4379 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 6243 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵 ↔ 𝐹:{𝐴}⟶𝐵)) |
3 | fveq2 6412 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2864 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵)) |
5 | eqidd 2801 | . . . . 5 ⊢ (𝑎 = 𝐴 → 𝐹 = 𝐹) | |
6 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
7 | 6, 3 | opeq12d 4602 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
8 | 7 | sneqd 4381 | . . . . 5 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉} = {〈𝐴, (𝐹‘𝐴)〉}) |
9 | 5, 8 | eqeq12d 2815 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
10 | 4, 9 | anbi12d 625 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) |
11 | 2, 10 | bibi12d 337 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶𝐵 ↔ ((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉})) ↔ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})))) |
12 | vex 3389 | . . 3 ⊢ 𝑎 ∈ V | |
13 | 12 | fsn2 6631 | . 2 ⊢ (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉})) |
14 | 11, 13 | vtoclg 3454 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {csn 4369 〈cop 4375 ⟶wf 6098 ‘cfv 6102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 |
This theorem is referenced by: fsnex 6767 pt1hmeo 21937 k0004val0 39229 difmapsn 40151 |
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