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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 4666 | . . . . 5 ⊢ 𝐴 ∈ {𝐴} |
3 | ffvelcdm 7100 | . . . . 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 6089 | . . . . . . . . 9 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
9 | fndm 6671 | . . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴}) | |
10 | 9 | imaeq2d 6079 | . . . . . . . . 9 ⊢ (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴})) |
11 | 8, 10 | eqtr3id 2788 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴})) |
12 | fnsnfv 6987 | . . . . . . . . 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 2777 | . . . . . . 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 4812 | . . . 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 7154 | . . 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 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 {csn 4630 〈cop 4636 dom cdm 5688 ran crn 5689 “ cima 5691 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 |
This theorem is referenced by: fsn2g 7157 fnressn 7177 fressnfv 7179 mapsnconst 8930 elixpsn 8975 en1 9062 mat1dimelbas 22492 0spth 30154 wlkl0 30395 ldepsnlinclem1 48350 ldepsnlinclem2 48351 0aryfvalel 48483 1arymaptf1 48491 |
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