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Mirrors > Home > ILE Home > Th. List > fsng | GIF version |
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.) |
Ref | Expression |
---|---|
fsng | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3581 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 5319 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏})) |
3 | opeq1 3752 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
4 | 3 | sneqd 3583 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
5 | 4 | eqeq2d 2176 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, 𝑏〉} ↔ 𝐹 = {〈𝐴, 𝑏〉})) |
6 | 2, 5 | bibi12d 234 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {〈𝑎, 𝑏〉}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {〈𝐴, 𝑏〉}))) |
7 | sneq 3581 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
8 | feq3 5316 | . . . 4 ⊢ ({𝑏} = {𝐵} → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵})) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵})) |
10 | opeq2 3753 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
11 | 10 | sneqd 3583 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
12 | 11 | eqeq2d 2176 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, 𝑏〉} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
13 | 9, 12 | bibi12d 234 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {〈𝐴, 𝑏〉}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉}))) |
14 | vex 2724 | . . 3 ⊢ 𝑎 ∈ V | |
15 | vex 2724 | . . 3 ⊢ 𝑏 ∈ V | |
16 | 14, 15 | fsn 5651 | . 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {〈𝑎, 𝑏〉}) |
17 | 6, 13, 16 | vtocl2g 2785 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 {csn 3570 〈cop 3573 ⟶wf 5178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 |
This theorem is referenced by: fsn2 5653 xpsng 5654 ftpg 5663 fseq1p1m1 10019 |
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