| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fsn2 | 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 NM, 19-May-2004.) |
| Ref | Expression |
|---|---|
| fsn2.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| fsn2 | ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fsn2.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
| 2 | 1 | snid 4662 | . . . . 5 ⊢ 𝐴 ∈ {𝐴} |
| 3 | ffvelcdm 7101 | . . . . 5 ⊢ ((𝐹:{𝐴}⟶𝐵 ∧ 𝐴 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) | |
| 4 | 2, 3 | mpan2 691 | . . . 4 ⊢ (𝐹:{𝐴}⟶𝐵 → (𝐹‘𝐴) ∈ 𝐵) |
| 5 | ffn 6736 | . . . . 5 ⊢ (𝐹:{𝐴}⟶𝐵 → 𝐹 Fn {𝐴}) | |
| 6 | dffn3 6748 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹) | |
| 7 | 6 | biimpi 216 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹) |
| 8 | imadmrn 6088 | . . . . . . . . 9 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
| 9 | fndm 6671 | . . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴}) | |
| 10 | 9 | imaeq2d 6078 | . . . . . . . . 9 ⊢ (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴})) |
| 11 | 8, 10 | eqtr3id 2791 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴})) |
| 12 | fnsnfv 6988 | . . . . . . . . 9 ⊢ ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
| 13 | 2, 12 | mpan2 691 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
| 14 | 11, 13 | eqtr4d 2780 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹‘𝐴)}) |
| 15 | 14 | feq3d 6723 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → (𝐹:{𝐴}⟶ran 𝐹 ↔ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
| 16 | 7, 15 | mpbid 232 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) |
| 17 | 5, 16 | syl 17 | . . . 4 ⊢ (𝐹:{𝐴}⟶𝐵 → 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) |
| 18 | 4, 17 | jca 511 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 → ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
| 19 | snssi 4808 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → {(𝐹‘𝐴)} ⊆ 𝐵) | |
| 20 | fss 6752 | . . . . 5 ⊢ ((𝐹:{𝐴}⟶{(𝐹‘𝐴)} ∧ {(𝐹‘𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵) | |
| 21 | 20 | ancoms 458 | . . . 4 ⊢ (({(𝐹‘𝐴)} ⊆ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) → 𝐹:{𝐴}⟶𝐵) |
| 22 | 19, 21 | sylan 580 | . . 3 ⊢ (((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) → 𝐹:{𝐴}⟶𝐵) |
| 23 | 18, 22 | impbii 209 | . 2 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
| 24 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
| 25 | 1, 24 | fsn 7155 | . . 3 ⊢ (𝐹:{𝐴}⟶{(𝐹‘𝐴)} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
| 26 | 25 | anbi2i 623 | . 2 ⊢ (((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
| 27 | 23, 26 | bitri 275 | 1 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 {csn 4626 〈cop 4632 dom cdm 5685 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: fsn2g 7158 fnressn 7178 fressnfv 7180 mapsnconst 8932 elixpsn 8977 en1 9064 mat1dimelbas 22477 0spth 30145 wlkl0 30386 ldepsnlinclem1 48422 ldepsnlinclem2 48423 0aryfvalel 48555 1arymaptf1 48563 |
| Copyright terms: Public domain | W3C validator |