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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 4212 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏}) | |
2 | 1 | fneq2d 5896 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏})) |
3 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | fveq2 6103 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
5 | 3, 4 | opeq12d 4348 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
6 | 5 | preq1d 4218 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
7 | 6 | eqeq2d 2620 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉})) |
8 | 2, 7 | bibi12d 334 | . 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 4213 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵}) | |
10 | 9 | fneq2d 5896 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵})) |
11 | id 22 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
12 | fveq2 6103 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
13 | 11, 12 | opeq12d 4348 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝑏, (𝐹‘𝑏)〉 = 〈𝐵, (𝐹‘𝐵)〉) |
14 | 13 | preq2d 4219 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉}) |
15 | 14 | eqeq2d 2620 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
16 | 10, 15 | bibi12d 334 | . 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 3176 | . . 3 ⊢ 𝑎 ∈ V | |
18 | vex 3176 | . . 3 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | fnprb 6377 | . 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
20 | 8, 16, 19 | vtocl2g 3243 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cpr 4127 〈cop 4131 Fn wfn 5799 ‘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-8 1979 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-pow 4769 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-csb 3500 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-mpt 4645 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: fpr2g 6380 |
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