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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 3602 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 5348 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏})) |
3 | opeq1 3776 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
4 | 3 | sneqd 3604 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
5 | 4 | eqeq2d 2189 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, 𝑏〉} ↔ 𝐹 = {〈𝐴, 𝑏〉})) |
6 | 2, 5 | bibi12d 235 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {〈𝑎, 𝑏〉}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {〈𝐴, 𝑏〉}))) |
7 | sneq 3602 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
8 | feq3 5345 | . . . 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 3777 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
11 | 10 | sneqd 3604 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
12 | 11 | eqeq2d 2189 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, 𝑏〉} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
13 | 9, 12 | bibi12d 235 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {〈𝐴, 𝑏〉}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉}))) |
14 | vex 2740 | . . 3 ⊢ 𝑎 ∈ V | |
15 | vex 2740 | . . 3 ⊢ 𝑏 ∈ V | |
16 | 14, 15 | fsn 5683 | . 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {〈𝑎, 𝑏〉}) |
17 | 6, 13, 16 | vtocl2g 2801 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {csn 3591 〈cop 3594 ⟶wf 5207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 |
This theorem is referenced by: fsn2 5685 xpsng 5686 ftpg 5695 fseq1p1m1 10067 intopsn 12665 grp1inv 12853 |
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