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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 4641 | . . 3 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 6723 | . 2 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵 ↔ 𝐹:{𝐴}⟶𝐵)) |
3 | fveq2 6907 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2824 | . . 3 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵)) |
5 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
6 | 5, 3 | opeq12d 4886 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
7 | 6 | sneqd 4643 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉} = {〈𝐴, (𝐹‘𝐴)〉}) |
8 | 7 | eqeq2d 2746 | . . 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 3482 | . . 3 ⊢ 𝑎 ∈ V | |
11 | 10 | fsn2 7156 | . 2 ⊢ (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉})) |
12 | 2, 9, 11 | vtoclbg 3557 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 ⟶wf 6559 ‘cfv 6563 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: fsnex 7303 pt1hmeo 23830 k0004val0 44144 difmapsn 45155 fsetsniunop 46999 f1sn2g 48681 |
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