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 4155 | . . . . 5 ⊢ 𝐴 ∈ {𝐴} |
3 | ffvelrn 6265 | . . . . 5 ⊢ ((𝐹:{𝐴}⟶𝐵 ∧ 𝐴 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) | |
4 | 2, 3 | mpan2 703 | . . . 4 ⊢ (𝐹:{𝐴}⟶𝐵 → (𝐹‘𝐴) ∈ 𝐵) |
5 | ffn 5958 | . . . . 5 ⊢ (𝐹:{𝐴}⟶𝐵 → 𝐹 Fn {𝐴}) | |
6 | dffn3 5967 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹) | |
7 | 6 | biimpi 205 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹) |
8 | imadmrn 5395 | . . . . . . . . 9 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
9 | fndm 5904 | . . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴}) | |
10 | 9 | imaeq2d 5385 | . . . . . . . . 9 ⊢ (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴})) |
11 | 8, 10 | syl5eqr 2658 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴})) |
12 | fnsnfv 6168 | . . . . . . . . 9 ⊢ ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
13 | 2, 12 | mpan2 703 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
14 | 11, 13 | eqtr4d 2647 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹‘𝐴)}) |
15 | 14 | feq3d 5945 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → (𝐹:{𝐴}⟶ran 𝐹 ↔ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
16 | 7, 15 | mpbid 221 | . . . . 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 553 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 → ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
19 | snssi 4280 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → {(𝐹‘𝐴)} ⊆ 𝐵) | |
20 | fss 5969 | . . . . 5 ⊢ ((𝐹:{𝐴}⟶{(𝐹‘𝐴)} ∧ {(𝐹‘𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵) | |
21 | 20 | ancoms 468 | . . . 4 ⊢ (({(𝐹‘𝐴)} ⊆ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) → 𝐹:{𝐴}⟶𝐵) |
22 | 19, 21 | sylan 487 | . . 3 ⊢ (((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) → 𝐹:{𝐴}⟶𝐵) |
23 | 18, 22 | impbii 198 | . 2 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
24 | fvex 6113 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
25 | 1, 24 | fsn 6308 | . . 3 ⊢ (𝐹:{𝐴}⟶{(𝐹‘𝐴)} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
26 | 25 | anbi2i 726 | . 2 ⊢ (((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
27 | 23, 26 | bitri 263 | 1 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 〈cop 4131 dom cdm 5038 ran crn 5039 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: fsn2g 6311 fnressn 6330 fressnfv 6332 mapsnconst 7789 elixpsn 7833 en1 7909 mat1dimelbas 20096 0spth 26101 0spth-av 41294 ldepsnlinclem1 42088 ldepsnlinclem2 42089 |
Copyright terms: Public domain | W3C validator |