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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 4672 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏}) | |
2 | 1 | fneq2d 6450 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏})) |
3 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | fveq2 6673 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
5 | 3, 4 | opeq12d 4814 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, (𝐹‘𝑎)〉 = 〈𝐴, (𝐹‘𝐴)〉) |
6 | 5 | preq1d 4678 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
7 | 6 | eqeq2d 2835 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉})) |
8 | 2, 7 | bibi12d 348 | . 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 4673 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵}) | |
10 | 9 | fneq2d 6450 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵})) |
11 | id 22 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
12 | fveq2 6673 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
13 | 11, 12 | opeq12d 4814 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝑏, (𝐹‘𝑏)〉 = 〈𝐵, (𝐹‘𝐵)〉) |
14 | 13 | preq2d 4679 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉}) |
15 | 14 | eqeq2d 2835 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝑏, (𝐹‘𝑏)〉} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉, 〈𝐵, (𝐹‘𝐵)〉})) |
16 | 10, 15 | bibi12d 348 | . 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 3500 | . . 3 ⊢ 𝑎 ∈ V | |
18 | vex 3500 | . . 3 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | fnprb 6974 | . 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {〈𝑎, (𝐹‘𝑎)〉, 〈𝑏, (𝐹‘𝑏)〉}) |
20 | 8, 16, 19 | vtocl2g 3575 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {cpr 4572 〈cop 4576 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: fpr2g 6977 |
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