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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 4738 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏}) | |
2 | 1 | fneq2d 6663 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏})) |
3 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | fveq2 6907 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
5 | 3, 4 | opeq12d 4886 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
6 | 5 | preq1d 4744 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
7 | 6 | eqeq2d 2746 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉})) |
8 | 2, 7 | bibi12d 345 | . 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 4739 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵}) | |
10 | 9 | fneq2d 6663 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵})) |
11 | id 22 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
12 | fveq2 6907 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
13 | 11, 12 | opeq12d 4886 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝑏, (𝐹‘𝑏)〉 = 〈𝐵, (𝐹‘𝐵)〉) |
14 | 13 | preq2d 4745 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉}) |
15 | 14 | eqeq2d 2746 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
16 | 10, 15 | bibi12d 345 | . 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 3482 | . . 3 ⊢ 𝑎 ∈ V | |
18 | vex 3482 | . . 3 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | fnprb 7228 | . 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
20 | 8, 16, 19 | vtocl2g 3574 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cpr 4633 〈cop 4637 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: fpr2g 7231 |
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