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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 4570 | . . 3 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 6494 | . 2 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵 ↔ 𝐹:{𝐴}⟶𝐵)) |
3 | fveq2 6664 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2897 | . . 3 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵)) |
5 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
6 | 5, 3 | opeq12d 4804 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
7 | 6 | sneqd 4572 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉} = {〈𝐴, (𝐹‘𝐴)〉}) |
8 | 7 | eqeq2d 2832 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
9 | 4, 8 | anbi12d 632 | . 2 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) |
10 | vex 3497 | . . 3 ⊢ 𝑎 ∈ V | |
11 | 10 | fsn2 6892 | . 2 ⊢ (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉})) |
12 | 2, 9, 11 | vtoclbg 3568 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4560 〈cop 4566 ⟶wf 6345 ‘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: fsnex 7033 pt1hmeo 22408 k0004val0 40497 difmapsn 41468 |
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