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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 4539 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏}) | |
2 | 1 | fneq2d 6277 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏})) |
3 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | fveq2 6496 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
5 | 3, 4 | opeq12d 4681 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
6 | 5 | preq1d 4545 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
7 | 6 | eqeq2d 2782 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉})) |
8 | 2, 7 | bibi12d 338 | . 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 4540 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵}) | |
10 | 9 | fneq2d 6277 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵})) |
11 | id 22 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
12 | fveq2 6496 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
13 | 11, 12 | opeq12d 4681 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝑏, (𝐹‘𝑏)〉 = 〈𝐵, (𝐹‘𝐵)〉) |
14 | 13 | preq2d 4546 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉}) |
15 | 14 | eqeq2d 2782 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
16 | 10, 15 | bibi12d 338 | . 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 3412 | . . 3 ⊢ 𝑎 ∈ V | |
18 | vex 3412 | . . 3 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | fnprb 6795 | . 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
20 | 8, 16, 19 | vtocl2g 3484 | 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 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {cpr 4437 〈cop 4441 Fn wfn 6180 ‘cfv 6185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 |
This theorem is referenced by: fpr2g 6798 |
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