Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fnpr2g | Structured version Visualization version GIF version |
Description: A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
Ref | Expression |
---|---|
fnpr2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 4268 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏}) | |
2 | 1 | fneq2d 5982 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏})) |
3 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | fveq2 6191 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
5 | 3, 4 | opeq12d 4410 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
6 | 5 | preq1d 4274 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
7 | 6 | eqeq2d 2632 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉})) |
8 | 2, 7 | bibi12d 335 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉}) ↔ (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}))) |
9 | preq2 4269 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵}) | |
10 | 9 | fneq2d 5982 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵})) |
11 | id 22 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
12 | fveq2 6191 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
13 | 11, 12 | opeq12d 4410 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝑏, (𝐹‘𝑏)〉 = 〈𝐵, (𝐹‘𝐵)〉) |
14 | 13 | preq2d 4275 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉}) |
15 | 14 | eqeq2d 2632 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
16 | 10, 15 | bibi12d 335 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}) ↔ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉}))) |
17 | vex 3203 | . . 3 ⊢ 𝑎 ∈ V | |
18 | vex 3203 | . . 3 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | fnprb 6472 | . 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
20 | 8, 16, 19 | vtocl2g 3270 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cpr 4179 〈cop 4183 Fn wfn 5883 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: fpr2g 6475 |
Copyright terms: Public domain | W3C validator |